Methods, apparatus and systems for channel estimation and simultaneous beamforming training for multi-input multi-output (MIMO) communications

ABSTRACT

Methods, apparatuses, and systems to enable a channel estimation of more than one channel for Multi-Input Multi-Output (MIMO) communications in a wireless network is provided. The method includes determining a first set of complex numbers comprising first channel estimation signal associated with a first channel, determining a second set of complex numbers comprising second channel estimation signal associated with a second channel, and transmitting the first set of complex numbers and the second set of complex numbers via a physical layer (PHY) frame, wherein the second set of complex numbers are complex conjugates of the first set of complex numbers.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a Continuation of U.S. patent application Ser. No.15/758,921, filed Mar. 9, 2018, which issued as U.S. Pat. No. 11,038,720on Jun. 15, 2021 and is the National Stage Application filed under 35U.S.C. 371 of International Application No. PCT/US2016/050399 filed Sep.6, 2016 and claims priority to U.S. Provisional Patent Application No.62/216,604 filed Sep. 10, 2015 and to U.S. Provisional PatentApplication No. 62/365,043, filed Jul. 21, 2016, the disclosures ofwhich are incorporated herein by reference in their entirety.

FIELD

The present invention relates to the field of wireless communicationsand, more particularly, to methods, apparatus and systems for performingchannel estimation and simultaneous beamforming training for MIMOcommunications.

BACKGROUND

A Wireless Local Area Network (WLAN) may provide wireless communicationservices. A WLAN may have a plurality of mode, e.g., an infrastructureand ad-hoc mode. In ad-hoc mode, one or more STAs transmit directly inpeer-to-peer (P2P). In infrastructure mode, one or more stations (STAs)communicate through an Access Point (AP) that serves as a bridge toother networks (such as Internet or Local Area Network). A WLANoperating in infrastructure mode may provide MIMO communications, suchas Multi-User MIMO (MU-MIMO).

WLAN systems which support multiple channels, and channel widths, suchas 802.11n, 802.11ac, 802.11af, and 802.11ah, include a channel which isdesignated as a primary channel. To improve spectral efficiency,802.11ac has introduced a concept for downlink MU-MIMO transmission tomultiple STAs in the same symbol's time frame, e.g., during a downlinkOrthogonal Frequency Division Multiplexing (OFDM) symbol. The potentialfor use of downlink MU-MIMO has been also considered for 802.11ah. It isimportant to note that downlink MU-MIMO uses the same symbol timing tomultiple STAs interference of waveform transmissions to multiple STAs.All STAs involved in MU-MIMO transmission with an AP may use the samechannel or band. This limits operating bandwidth to the smallest channelbandwidth that is supported by the STAs which are included in theMU-MIMO transmission with the AP. In order to solve such limitations,the present application proposes methods, apparatus, and systems forchannel estimation and simultaneous beamforming training for MIMOcommunications.

SUMMARY

Methods, apparatuses, and systems to enable a channel estimation of morethan one channel for MIMO communications. Also methods, apparatuses, andsystems to perform simultaneous beamforming training in a wirelessnetwork are provided.

A representative method for an electronic device to transmit informationfor channel estimation for more than one channel for MIMO communicationsin a wireless network includes determining a first set of complexnumbers comprising first channel estimation signal associated with afirst channel, determining a second set of complex numbers comprisingsecond channel estimation signal associated with a second channel, andtransmitting the first set of complex numbers and the second set ofcomplex numbers via a physical layer (PHY) frame, wherein the second setof complex numbers are complex conjugates of the first set of complexnumbers.

A representative method for an electronic device to perform channelestimation of more than one channel for MIMO communications in awireless network includes receiving a first set of complex numberscomprising a first channel estimation signal and a second set of complexnumbers comprising a second channel estimation signal, and determining afirst channel based on the first channel estimation signal and a secondchannel based on the second channel estimation signal, wherein the firstset of complex numbers and the second set of complex numbers arereceived via a PHY frame, and wherein the second set of complex numbersare complex conjugates of the first set of complex numbers.

A representative method for an electronic device to transmit beamformingfor MIMO communications in a wireless network includes performing aSector Level Sweep (SLS) and Beam Refinement Protocol (BRP) to identifya beam for all of one or more stations (STAs), identifying a first STAand a second STA disposed in adjacent beams, transmitting a multi-usertransmission to the first STA and the second STA using a PHY frameincluding information on a first stream corresponding to a first channeland information on a second stream corresponding to a second channel,and simultaneously transmitting user specific information to the firstSTA and to the second STA in the PHY frame, wherein the PHY frameincludes a legacy short training field (STF), a long training field(LTF), and a signal (SIG) on both the first stream and the secondstream, and wherein the PHY frame includes a common header includinginformation on the one or more STAs and information associating thefirst stream with the first STA and associating the second stream withthe second STA.

A representative apparatus includes an electronic device configured totransmit information for channel estimation for more than one channelfor MIMO communications in a wireless network. The electronic deviceincludes a processor configured to (1) determine a first set of complexnumbers comprising first channel estimation signal associated with afirst channel and (2) determine a second set of complex numberscomprising second channel estimation signal associated with a secondchannel, and a transmitter configured to the first set of complexnumbers and the second set of complex numbers via a PHY frame, whereinthe second set of complex numbers are complex conjugates of the firstset of complex numbers.

A representative apparatus includes an electronic device configured toperform channel estimation of more than one channel for MIMOcommunications in a wireless network. The electronic device includes areceiver configured to receive a first set of complex numbers comprisinga first channel estimation signal and a second set of complex numberscomprising a second channel estimation signal, and a processorconfigured to determine a first channel based on the first channelestimation signal and a second channel based on the second channelestimation signal, wherein the first set of complex numbers and thesecond set of complex numbers are received via a PHY frame, and whereinthe second set of complex numbers are complex conjugates of the firstset of complex numbers.

A representative apparatus includes an electronic device configured toperform transmit beamforming for MIMO communications in a wirelessnetwork. The electronic device includes a processor configured toperform a Sector Level Sweep (SLS) and Beam Refinement Protocol (BRP) toidentify a beam for all of one or more stations (STAs), identify a firstSTA and a second STA disposed in adjacent beams, transmit a multi-usertransmission to the first STA and the second STA using a PHY frameincluding information on a first stream corresponding to a first channeland information on a second stream corresponding to a second channel,and simultaneously transmit user specific information to the first STAand to the second STA in the PHY frame, wherein the PHY frame includes alegacy short training field (STF), a long training field (LTF), and asignal (SIG) on both the first stream and the second stream, and whereinthe PHY frame includes a common header including information on the oneor more STAs and information associating the first stream with the firstSTA and associating the second stream with the second STA.

BRIEF DESCRIPTION OF THE DRAWINGS

A more detailed understanding may be had from the Detailed Descriptionbelow, given by way of example in conjunction with drawings appendedhereto. Figures in such drawings, like the detailed description, areexamples. As such, the Figures and the detailed description are not tobe considered limiting, and other equally effective examples arepossible and likely.

Furthermore, like reference numerals in the Figures indicate likeelements, and wherein:

FIG. 1 is a system diagram illustrating an example communications systemin which one or more disclosed embodiments may be implemented;

FIG. 2 is a system diagram illustrating an example wirelesstransmit/receive unit (WTRU) that may be used within the communicationssystem illustrated in FIG. 1 ;

FIG. 3 is a system diagram illustrating an example radio access networkand another example core network that may be used within thecommunications system illustrated in FIG. 1 ;

FIG. 4 is a system diagram illustrating another example radio accessnetwork and another example core network that may be used within thecommunications system illustrated in FIG. 1 ;

FIG. 5 is a system diagram illustrating a further example radio accessnetwork and a further example core network that may be used within thecommunications system illustrated in FIG. 1 ;

FIG. 6 illustrates different PHY frame types according to an embodiment;

FIG. 7 illustrates a preamble structure according to an embodiment;

FIG. 8 illustrates a beamforming training field structure according toan embodiment;

FIG. 9 illustrates a flowchart of performing MIMO transmissions;

FIG. 10 illustrates Channel Estimation Fields (CEFs) for multiplestreams of a single carrier transmission according to an embodiment;

FIG. 11 illustrates autocorrelation zones and cross correlation zones ofsequences s_(uv) and s_(vu) according to an embodiment;

FIG. 12 illustrates CEFs for multiple streams according to anembodiment;

FIG. 13 illustrates reduced CEFs for multiple streams according to anembodiment;

FIG. 14 illustrates interoperability of 802.11ad and 802.11 ay systemsaccording to an embodiment;

FIG. 15 illustrates channel estimation results of the 802.11ad receiver;

FIG. 16 illustrates channel estimation results of the 802.11ay receiveraccording to an embodiment;

FIG. 17 illustrates a header for an 802.11 ay packet according to anembodiment;

FIG. 18 illustrates CEFs for two streams, according to an embodiment;

FIG. 19 illustrates an extra CEF according to an embodiment;

FIG. 20 illustrates CEFs for multiple streams according to anembodiment;

FIG. 21 illustrates a preamble and PHY header for MIMO transmissions,according to an embodiment;

FIG. 22 illustrates a preamble and PHY header for MIMO transmissionsaccording to another embodiment;

FIG. 23 illustrates a preamble and PHY header for MIMO transmissionsaccording to another embodiment;

FIG. 24 illustrates a preamble and PHY header according to yet anotherembodiment;

FIG. 25 illustrates a preamble and PHY header according to anembodiment;

FIG. 26 illustrates a preamble and PHY header according to anotherembodiment;

FIG. 27 illustrates a MU transmission having a large degree ofseparation according to an embodiment;

FIG. 28 illustrates a frame structure for a MU transmission having alarge degree of separation according to an embodiment;

FIG. 29 illustrates a MU transmission having a small degree ofseparation according to an embodiment;

FIG. 30 illustrates a frame structure for a MU transmission having asmall degree of separation according to an embodiment;

FIG. 31 illustrates receive/transmit (R/T) sequences for two steams;

FIG. 32 illustrates Zero autocorrelation zones for arguments (a) and (b)and zero cross correlation zones for argument (c), according to anembodiment;

FIG. 33 illustrates beam training fields using complex conjugates of R/Tsequences according to an embodiment;

FIG. 34 illustrates beam training fields using R/T sequences based on anorthogonal V matrix according to an embodiment;

FIG. 35 illustrates a PHY header for a combination of sequential andparallel beamforming training according to an embodiment;

FIG. 36 illustrates auto detection of simultaneous beamforming trainingfor a LP-SC/SC PHY header according to an embodiment;

FIG. 37 illustrates implicit detection of the beams to be trained byusing different sequences in the AGC fields;

FIG. 38 illustrates an exemplary flowchart of transmitting informationfor channel estimation for more than one channel for MIMO communicationsin a wireless network;

FIG. 39 illustrates another exemplary flowchart of performing channelestimation of more than one channel for MIMO communications in awireless network; and

FIG. 40 illustrates another exemplary flowchart of performing transmitbeamforming for MIMO communications in a wireless network.

DETAILED DESCRIPTION

A detailed description of illustrative embodiments may now be describedwith reference to the figures. However, while the present invention maybe described in connection with representative embodiments, it is notlimited thereto and it is to be understood that other embodiments may beused or modifications and additions may be made to the describedembodiments for performing the same function of the present inventionwithout deviating therefrom.

Although the representative embodiments are generally shown hereafterusing wireless network architectures, any number of different networkarchitectures may be used including networks with wired componentsand/or wireless components, for example.

FIG. 1 is a diagram illustrating an example communications system 100 inwhich one or more disclosed embodiments may be implemented. Thecommunications system 100 may be a multiple access system that providescontent, such as voice, data, video, messaging, broadcast, etc., tomultiple wireless users. The communications system 100 may enablemultiple wireless users to access such content through the sharing ofsystem resources, including wireless bandwidth. For example, thecommunications systems 100 may employ one or more channel accessmethods, such as code division multiple access (CDMA), time divisionmultiple access (TDMA), frequency division multiple access (FDMA),orthogonal FDMA (OFDMA), single-carrier FDMA (SC-FDMA), and the like.

As shown in FIG. 1 , the communications system 100 may includeelectronic devices such as wireless transmit/receive units (WTRUs) 102a, 102 b, 102 c, 102 d, a radio access network (RAN) 104, a core network106/107/109, a public switched telephone network (PSTN) 108, theInternet 110, and other networks 112, though it will be appreciated thatthe disclosed embodiments contemplate any number of WTRUs, basestations, networks, and/or network elements. Each of the WTRUs 102 a,102 b, 102 c, 102 d may be any type of device configured to operateand/or communicate in a wireless environment. By way of example, theWTRUs 102 a, 102 b, 102 c, 102 d, which may be referred to as a“station” and/or a “STA”, may be configured to transmit and/or receivewireless signals and may include user equipment (UE), a mobile station,a fixed or mobile subscriber unit, a pager, a cellular telephone, apersonal digital assistant (PDA), a smartphone, a laptop, a netbook, apersonal computer, a wireless sensor, consumer electronics, and thelike. The WTRU 102 a, 102 b, 102 c and 102 d is interchangeably referredto as a UE.

The communications systems 100 may also include electronic devices suchas a base station 114 a and/or a base station 114 b. Each of the basestations 114 a, 114 b may be any type of device configured to wirelesslyinterface with at least one of the WTRUs 102 a, 102 b, 102 c, 102 d tofacilitate access to one or more communication networks, such as thecore network 106/107/109, the Internet 110, and/or the other networks112. By way of example, the base stations 114 a, 114 b may be a basetransceiver station (BTS), a Node-B, an eNode B, a Home Node B, a HomeeNode B, a site controller, an access point (AP), a wireless router, andthe like. While the base stations 114 a, 114 b are each depicted as asingle element, it will be appreciated that the base stations 114 a, 114b may include any number of interconnected base stations and/or networkelements.

The base station 114 a may be part of the RAN 103/104/105, which mayalso include other base stations and/or network elements (not shown),such as a base station controller (BSC), a radio network controller(RNC), relay nodes, etc. The base station 114 a and/or the base station114 b may be configured to transmit and/or receive wireless signalswithin a particular geographic region, which may be referred to as acell (not shown). The cell may further be divided into cell sectors. Forexample, the cell associated with the base station 114 a may be dividedinto three sectors. Thus, in one embodiment, the base station 114 a mayinclude three transceivers, i.e., one for each sector of the cell. Inanother embodiments, the base station 114 a may employ MIMO technologyand may utilize multiple transceivers for each sector of the cell.

The base stations 114 a, 114 b may communicate with one or more of theWTRUs 102 a, 102 b, 102 c, 102 d over an air interface 115/116/117,which may be any suitable wireless communication link (e.g., radiofrequency (RF), microwave, infrared (IR), ultraviolet (UV), visiblelight, etc.). The air interface 115/116/117 may be established using anysuitable radio access technology (RAT).

More specifically, as noted above, the communications system 100 may bea multiple access system and may employ one or more channel accessschemes, such as CDMA, TDMA, FDMA, OFDMA, SC-FDMA, and the like. Forexample, the base station 114 a in the RAN 103/104/105 and the WTRUs 102a, 102 b, 102 c may implement a radio technology such as UniversalMobile Telecommunications System (UMTS) Terrestrial Radio Access (UTRA),which may establish the air interface 115/116/117 using wideband CDMA(WCDMA). WCDMA may include communication protocols such as High-SpeedPacket Access (HSPA) and/or Evolved HSPA (HSPA+). HSPA may includeHigh-Speed Downlink (DL) Packet Access (HSDPA) and/or High-Speed ULPacket Access (HSUPA).

In another embodiment, the base station 114 a and the WTRUs 102 a, 102b, 102 c may implement a radio technology such as Evolved UMTSTerrestrial Radio Access (E-UTRA), which may establish the air interface115/116/117 using Long Term Evolution (LTE) and/or LTE-Advanced (LTE-A).

In other embodiments, the base station 114 a and the WTRUs 102 a, 102 b,102 c may implement radio technologies such as IEEE 802.11 (i.e.,Wireless Fidelity (WiFi), IEEE 802.16 (i.e., Worldwide Interoperabilityfor Microwave Access (WiMAX)), CDMA2000, CDMA2000 1×, CDMA2000 EV-DO,Interim Standard 2000 (IS-2000), Interim Standard 95 (IS-95), InterimStandard 856 (IS-856), Global System for Mobile communications (GSM),Enhanced Data rates for GSM Evolution (EDGE), GSM EDGE (GERAN), and thelike.

The base station 114 b in FIG. 1 may be a wireless router, Home Node B,Home eNode B, or access point, for example, and may utilize any suitableRAT for facilitating wireless connectivity in a localized area, such asa place of business, a home, a vehicle, a campus, and the like. In oneembodiment, the base station 114 b and the WTRUs 102 c, 102 d mayimplement a radio technology such as IEEE 802.11 to establish a WLAN. Inanother embodiments, the base station 114 b and the WTRUs 102 c, 102 dmay implement a radio technology such as IEEE 802.15 to establish awireless personal area network (WPAN). In yet another embodiments, thebase station 114 b and the WTRUs 102 c, 102 d may utilize acellular-based RAT (e.g., WCDMA, CDMA2000, GSM, LTE, LTE-A, etc.) toestablish a picocell or femtocell. As shown in FIG. 1 , the base station114 b may have a direct connection to the Internet 110. Thus, the basestation 114 b may not be required to access the Internet 110 via thecore network 106/107/109.

The RAN 103/104/105 may be in communication with the core network106/107/109, which may be any type of network configured to providevoice, data, applications, and/or voice over internet protocol (VoIP)services to one or more of the WTRUs 102 a, 102 b, 102 c, 102 d. Forexample, the core network 106/107/109 may provide call control, billingservices, mobile location-based services, pre-paid calling, Internetconnectivity, video distribution, etc., and/or perform high-levelsecurity functions, such as user authentication. Although not shown inFIG. 1 , it will be appreciated that the RAN 103/104/105 and/or the corenetwork 106/107/109 may be in direct or indirect communication withother RANs that employ the same RAT as the RAN 103/104/105 or adifferent RAT. For example, in addition to being connected to the RAN103/104/105, which may be utilizing an E-UTRA radio technology, the corenetwork 106/107/109 may also be in communication with another RAN (notshown) employing a GSM, UMTS, CDMA 2000, WiMAX, or WiFi radiotechnology.

The core network 106/107/109 may also serve as a gateway for the WTRUs102 a, 102 b, 102 c, 102 d to access the PSTN 108, the Internet 110,and/or the other networks 112. The PSTN 108 may include circuit-switchedtelephone networks that provide plain old telephone service (POTS). TheInternet 110 may include a global system of interconnected computernetworks and devices that use common communication protocols, such asthe transmission control protocol (TCP), user datagram protocol (UDP)and/or the internet protocol (IP) in the TCP/IP internet protocol suite.The networks 112 may include wired and/or wireless communicationsnetworks owned and/or operated by other service providers. For example,the networks 112 may include another core network connected to one ormore RANs, which may employ the same RAT as the RAN 103/104/105 or adifferent RAT.

Some or all of the WTRUs 102 a, 102 b, 102 c, 102 d in thecommunications system 100 may include multi-mode capabilities (e.g., theWTRUs 102 a, 102 b, 102 c, 102 d may include multiple transceivers forcommunicating with different wireless networks over different wirelesslinks). For example, the WTRU 102 c shown in FIG. 1 may be configured tocommunicate with the base station 114 a, which may employ acellular-based radio technology, and with the base station 114 b, whichmay employ an IEEE 802 radio technology.

FIG. 2 is a system diagram illustrating an example WTRU 102. As shown inFIG. 2 , the WTRU 102 may include a processor 118, a transceiver 120, atransmit/receive element 122, a speaker/microphone 124, a keypad 126, adisplay/touchpad 128, non-removable memory 130, removable memory 132, apower source 134, a global positioning system (GPS) chipset 136, and/orother peripherals 138, among others. It will be appreciated that theWTRU 102 may include any sub-combination of the foregoing elements whileremaining consistent with an embodiment.

The processor 118 may be a general purpose processor, a special purposeprocessor, a conventional processor, a digital signal processor (DSP), aplurality of microprocessors, one or more microprocessors in associationwith a DSP core, a controller, a microcontroller, Application SpecificIntegrated Circuits (ASICs), Field Programmable Gate Array (FPGAs)circuits, any other type of integrated circuit (IC), a state machine,and the like. The processor 118 may perform signal coding, dataprocessing, power control, input/output processing, and/or any otherfunctionality that enables the WTRU 102 to operate in a wirelessenvironment. The processor 118 may be coupled to the transceiver 120,which may be coupled to the transmit/receive element 122. While FIG. 2depicts the processor 118 and the transceiver 120 as separatecomponents, it will be appreciated that the processor 118 and thetransceiver 120 may be integrated together in an electronic package orchip.

The transmit/receive element 122 may be configured to transmit signalsto, or receive signals from, a base station (e.g., the base station 114a) over the air interface 115/116/117. For example, in one embodiment,the transmit/receive element 122 may be an antenna configured totransmit and/or receive RF signals. In another embodiments, thetransmit/receive element 122 may be an emitter/detector configured totransmit and/or receive IR, UV, or visible light signals, for example.In yet another embodiments, the transmit/receive element 122 may beconfigured to transmit and/or receive both RF and light signals. It willbe appreciated that the transmit/receive element 122 may be configuredto transmit and/or receive any combination of wireless signals.

Although the transmit/receive element 122 is depicted in FIG. 2 as asingle element, the WTRU 102 may include any number of transmit/receiveelements 122. More specifically, the WTRU 102 may employ MIMOtechnology. Thus, in one embodiment, the WTRU 102 may include two ormore transmit/receive elements 122 (e.g., multiple antennas) fortransmitting and receiving wireless signals over the air interface115/116/117.

The transceiver 120 may be configured to modulate the signals that areto be transmitted by the transmit/receive element 122 and to demodulatethe signals that are received by the transmit/receive element 122. Asnoted above, the WTRU 102 may have multi-mode capabilities. Thus, thetransceiver 120 may include multiple transceivers for enabling the WTRU102 to communicate via multiple RATs, such as UTRA and IEEE 802.11, forexample.

The processor 118 of the WTRU 102 may be coupled to, and may receiveuser input data from, the speaker/microphone 124, the keypad 126, and/orthe display/touchpad 128 (e.g., a liquid crystal display (LCD) displayunit or organic light-emitting diode (OLED) display unit). The processor118 may also output user data to the speaker/microphone 124, the keypad126, and/or the display/touchpad 128. In addition, the processor 118 mayaccess information from, and store data in, any type of suitable memory,such as the non-removable memory 130 and/or the removable memory 132.The non-removable memory 130 may include random-access memory (RAM),read-only memory (ROM), a hard disk, or any other type of memory storagedevice. The removable memory 132 may include a subscriber identitymodule (SIM) card, a memory stick, a secure digital (SD) memory card,and the like. In other embodiments, the processor 118 may accessinformation from, and store data in, memory that is not physicallylocated on the WTRU 102, such as on a server or a home computer (notshown).

The processor 118 may receive power from the power source 134, and maybe configured to distribute and/or control the power to the othercomponents in the WTRU 102. The power source 134 may be any suitabledevice for powering the WTRU 102. For example, the power source 134 mayinclude one or more dry cell batteries (e.g., nickel-cadmium (NiCd),nickel-zinc (NiZn), nickel metal hydride (NiMH), lithium-ion (Li-ion),etc.), solar cells, fuel cells, and the like.

The processor 118 may also be coupled to the GPS chipset 136, which maybe configured to provide location information (e.g., longitude andlatitude) regarding the current location of the WTRU 102. In additionto, or in lieu of, the information from the GPS chipset 136, the WTRU102 may receive location information over the air interface 115/116/117from a base station (e.g., base stations 114 a, 114 b) and/or determineits location based on the timing of the signals being received from twoor more nearby base stations. It will be appreciated that the WTRU 102may acquire location information by way of any suitablelocation-determination method while remaining consistent with anembodiment.

The processor 118 may further be coupled to other peripherals 138, whichmay include one or more software and/or hardware modules that provideadditional features, functionality and/or wired or wirelessconnectivity. For example, the peripherals 138 may include anaccelerometer, an e-compass, a satellite transceiver, a digital camera(for photographs and/or video), a universal serial bus (USB) port, avibration device, a television transceiver, a hands free headset, aBluetooth® module, a frequency modulated (FM) radio unit, a digitalmusic player, a media player, a video game player module, an Internetbrowser, and the like. In a case where the peripherals 138 includes oneor more sensors, the sensors may be one or more of a gyroscope, anaccelerometer; an orientation sensor, a proximity sensor, a temperaturesensor, a time sensor; a geolocation sensor; an altimeter, a lightsensor, a touch sensor, a magnetometer, a barometer, a gesture sensor,and/or a humidity sensor.

The WTRU 102 may include a full duplex radio for which transmission andreception of some or all of the signals (e.g., associated withparticular subframes for both the UL (e.g., for transmission) anddownlink (e.g., for reception) may be concurrent and/or simultaneous.The full duplex radio may include an interference management unit 139 toreduce and or substantially eliminate self-interference via eitherhardware (e.g., a choke) or signal processing via a processor (e.g., aseparate processor (not shown) or via processor 118).

FIG. 3 is a system diagram illustrating the RAN 103 and the core network106 according to another embodiments. As noted above, the RAN 103 mayemploy a UTRA radio technology to communicate with the WTRUs 102 a, 102b, 102 c over the air interface 115. The RAN 103 may also be incommunication with the core network 106. As shown in FIG. 3 , the RAN103 may include Node-Bs 140 a, 140 b, 140 c, which may each include oneor more transceivers for communicating with the WTRUs 102 a, 102 b, 102c over the air interface 115. The Node-Bs 140 a, 140 b, 140 c may eachbe associated with a particular cell (not shown) within the RAN 103. TheRAN 103 may also include RNCs 142 a, 142 b. It will be appreciated thatthe RAN 103 may include any number of Node-Bs and RNCs while remainingconsistent with an embodiment.

As shown in FIG. 3 , the Node-Bs 140 a, 140 b may be in communicationwith the RNC 142 a. Additionally, the Node-B 140 c may be incommunication with the RNC 142 b. The Node-Bs 140 a, 140 b, 140 c maycommunicate with the respective RNCs 142 a, 142 b via an Iub interface.The RNCs 142 a, 142 b may be in communication with one another via anIur interface. Each of the RNCs 142 a, 142 b may be configured tocontrol the respective Node-Bs 140 a, 140 b, 140 c to which it isconnected. In addition, each of the RNCs 142 a, 142 b may be configuredto carry out or support other functionality, such as outer loop powercontrol, load control, admission control, packet scheduling, handovercontrol, macrodiversity, security functions, data encryption, and thelike.

The core network 106 shown in FIG. 3 may include a media gateway (MGW)144, a mobile switching center (MSC) 146, a serving GPRS support node(SGSN) 148, and/or a gateway GPRS support node (GGSN) 150. While each ofthe foregoing elements are depicted as part of the core network 106, itwill be appreciated that any one of these elements may be owned and/oroperated by an entity other than the core network operator.

The RNC 142 a in the RAN 103 may be connected to the MSC 146 in the corenetwork 106 via an IuCS interface. The MSC 146 may be connected to theMGW 144. The MSC 146 and the MGW 144 may provide the WTRUs 102 a, 102 b,102 c with access to circuit-switched networks, such as the PSTN 108, tofacilitate communications between the WTRUs 102 a, 102 b, 102 c andtraditional land-line communications devices.

The RNC 142 a in the RAN 103 may also be connected to the SGSN 148 inthe core network 106 via an IuPS interface. The SGSN 148 may beconnected to the GGSN 150. The SGSN 148 and the GGSN 150 may provide theWTRUs 102 a, 102 b, 102 c with access to packet-switched networks, suchas the Internet 110, to facilitate communications between and the WTRUs102 a, 102 b, 102 c and IP-enabled devices.

As noted above, the core network 106 may also be connected to the othernetworks 112, which may include other wired and/or wireless networksthat are owned and/or operated by other service providers.

FIG. 4 is a system diagram illustrating the RAN 104 and the core network107 according to an embodiment. As noted above, the RAN 104 may employan E-UTRA radio technology to communicate with the WTRUs 102 a, 102 b,102 c over the air interface 116. The RAN 104 may also be incommunication with the core network 107.

The RAN 104 may include eNode-Bs 160 a, 160 b, 160 c, though it will beappreciated that the RAN 104 may include any number of eNode-Bs whileremaining consistent with an embodiment. The eNode-Bs 160 a, 160 b, 160c may each include one or more transceivers for communicating with theWTRUs 102 a, 102 b, 102 c over the air interface 116. In one embodiment,the eNode-Bs 160 a, 160 b, 160 c may implement MIMO technology. Thus,the eNode-B 160 a, for example, may use multiple antennas to transmitwireless signals to, and/or receive wireless signals from, the WTRU 102a.

Each of the eNode-Bs 160 a, 160 b, 160 c may be associated with aparticular cell (not shown) and may be configured to handle radioresource management decisions, handover decisions, scheduling of usersin the UL and/or DL, and the like. As shown in FIG. 4 , the eNode-Bs 160a, 160 b, 160 c may communicate with one another over an X2 interface.

The core network 107 shown in FIG. 4 may include a mobility managemententity (MME) 162, a serving gateway (SGW) 164, and a packet data network(PDN) gateway (or PGW) 166. While each of the foregoing elements aredepicted as part of the core network 107, it will be appreciated thatany of these elements may be owned and/or operated by an entity otherthan the core network operator.

The MME 162 may be connected to each of the eNode-Bs 162 a, 162 b, 162 cin the RAN 104 via an S1 interface and may serve as a control node. Forexample, the MME 162 may be responsible for authenticating users of theWTRUs 102 a, 102 b, 102 c, bearer activation/deactivation, selecting aparticular serving gateway during an initial attach of the WTRUs 102 a,102 b, 102 c, and the like. The MME 162 may provide a control planefunction for switching between the RAN 104 and other RANs (not shown)that employ other radio technologies, such as GSM and/or WCDMA.

The serving gateway 164 may be connected to each of the eNode Bs 160 a,160 b, 160 c in the RAN 104 via the S1 interface. The serving gateway164 may generally route and forward user data packets to/from the WTRUs102 a, 102 b, 102 c. The serving gateway 164 may perform otherfunctions, such as anchoring user planes during inter-eNode B handovers,triggering paging when DL data is available for the WTRUs 102 a, 102 b,102 c, managing and storing contexts of the WTRUs 102 a, 102 b, 102 c,and the like.

The serving gateway 164 may be connected to the PDN gateway 166, whichmay provide the WTRUs 102 a, 102 b, 102 c with access to packet-switchednetworks, such as the Internet 110, to facilitate communications betweenthe WTRUs 102 a, 102 b, 102 c and IP-enabled devices.

The core network 107 may facilitate communications with other networks.For example, the core network 107 may provide the WTRUs 102 a, 102 b,102 c with access to circuit-switched networks, such as the PSTN 108, tofacilitate communications between the WTRUs 102 a, 102 b, 102 c andtraditional land-line communications devices. For example, the corenetwork 107 may include, or may communicate with, an IP gateway (e.g.,an IP multimedia subsystem (IMS) server) that serves as an interfacebetween the core network 107 and the PSTN 108. In addition, the corenetwork 107 may provide the WTRUs 102 a, 102 b, 102 c with access to theother networks 112, which may include other wired and/or wirelessnetworks that are owned and/or operated by other service providers.

FIG. 5 is a system diagram illustrating the RAN 105 and the core network109 according to an embodiment. The RAN 105 may be an access servicenetwork (ASN) that employs IEEE 802.16 radio technology to communicatewith the WTRUs 102 a, 102 b, 102 c over the air interface 117. As willbe further discussed below, the communication links between thedifferent functional entities of the WTRUs 102 a, 102 b, 102 c, the RAN105, and the core network 109 may be defined as reference points.

As shown in FIG. 5 , the RAN 105 may include base stations 180 a, 180 b,180 c, and an ASN gateway 182, though it will be appreciated that theRAN 105 may include any number of base stations and ASN gateways whileremaining consistent with an embodiment. The base stations 180 a, 180 b,180 c may each be associated with a particular cell (not shown) in theRAN 105 and may each include one or more transceivers for communicatingwith the WTRUs 102 a, 102 b, 102 c over the air interface 117. In oneembodiment, the base stations 180 a, 180 b, 180 c may implement MIMOtechnology. The base station 180 a, for example, may use multipleantennas to transmit wireless signals to, and/or receive wirelesssignals from, the WTRU 102 a. The base stations 180 a, 180 b, 180 c mayalso provide mobility management functions, such as handoff triggering,tunnel establishment, radio resource management, traffic classification,quality of service (QoS) policy enforcement, and the like. The ASNgateway 182 may serve as a traffic aggregation point and may beresponsible for paging, caching of subscriber profiles, routing to thecore network 109, and the like.

The air interface 117 between the WTRUs 102 a, 102 b, 102 c and the RAN105 may be defined as an R1 reference point that implements the IEEE802.16 specification. In addition, each of the WTRUs 102 a, 102 b, 102 cmay establish a logical interface (not shown) with the core network 109.The logical interface between the WTRUs 102 a, 102 b, 102 c and the corenetwork 109 may be defined as an R2 reference point, which may be usedfor authentication, authorization, IP host configuration management,and/or mobility management.

The communication link between each of the base stations 180 a, 180 b,180 c may be defined as an R8 reference point that includes protocolsfor facilitating WTRU handovers and the transfer of data between basestations. The communication link between the base stations 180 a, 180 b,180 c and the ASN gateway 182 may be defined as an R6 reference point.The R6 reference point may include protocols for facilitating mobilitymanagement based on mobility events associated with each of the WTRUs102 a, 102 b, 100 c.

As shown in FIG. 5 , the RAN 105 may be connected to the core network109. The communication link between the RAN 105 and the core network 109may be defined as an R3 reference point that includes protocols forfacilitating data transfer and mobility management capabilities, forexample. The core network 109 may include a mobile IP home agent(MIP-HA) 184, an authentication, authorization, accounting (AAA) server186, and a gateway 188. While each of the foregoing elements aredepicted as part of the core network 109, it will be appreciated thatany of these elements may be owned and/or operated by an entity otherthan the core network operator.

The MIP-HA 184 may be responsible for IP address management, and mayenable the WTRUs 102 a, 102 b, 102 c to roam between different ASNsand/or different core networks. The MIP-HA 184 may provide the WTRUs 102a, 102 b, 102 c with access to packet-switched networks, such as theInternet 110, to facilitate communications between the WTRUs 102 a, 102b, 102 c and IP-enabled devices. The AAA server 186 may be responsiblefor user authentication and for supporting user services. The gateway188 may facilitate interworking with other networks. For example, thegateway 188 may provide the WTRUs 102 a, 102 b, 102 c with access tocircuit-switched networks, such as the PSTN 108, to facilitatecommunications between the WTRUs 102 a, 102 b, 102 c and traditionalland-line communications devices. The gateway 188 may provide the WTRUs102 a, 102 b, 102 c with access to the other networks 112, which mayinclude other wired and/or wireless networks that are owned and/oroperated by other service providers.

Although not shown in FIG. 5 , it will be appreciated that the RAN 105may be connected to other ASNs, other RANS (e.g., RANs 103 and/or 104)and/or the core network 109 may be connected to other core networks(e.g., core network 106 and/or 107. The communication link between theRAN 105 and the other ASNs may be defined as an R4 reference point,which may include protocols for coordinating the mobility of the WTRUs102 a, 102 b, 102 c between the RAN 105 and the other ASNs. Thecommunication link between the core network 109 and the other corenetworks may be defined as an R5 reference, which may include protocolsfor facilitating interworking between home core networks and visitedcore networks.

Although the WTRU is described in FIGS. 1-5 as a wireless terminal, itis contemplated that in certain representative embodiments that such aterminal may use (e.g., temporarily or permanently) wired communicationinterfaces with the communication network.

In representative embodiments, the other network 112 may be a WLAN.

A WLAN in Infrastructure Basic Service Set (BSS) mode may have an AP forthe BSS and one or more STAs associated with the AP. The AP may have anaccess or an interface to a Distribution System (DS) or another type ofwired/wireless network that carries traffic in to and/or out of the BSS.Traffic to STAs that originates from outside the BSS may arrive throughthe AP and may be delivered to the STAs. Traffic originating from STAsto destinations outside the BSS may be sent to the AP to be delivered torespective destinations. Traffic between STAs within the BSS may be sentthrough the AP, for example, where the source STA may send traffic tothe AP and the AP may deliver the traffic to the destination STA. Thetraffic between STAs within a BSS may be considered and/or referred toas peer-to-peer traffic. The peer-to-peer traffic may be sent between(e.g., directly between) the source and destination STAs with a directlink setup (DLS). In certain representative embodiments, the DLS may usean 802.11e DLS or an 802.11z tunneled DLS (TDLS). A WLAN using anIndependent BSS (IBSS) mode may not have an AP, and the STAs (e.g., allof the STAs) within or using the IBSS may communicate directly with eachother. The IBSS mode of communication may sometimes be referred toherein as an “ad-hoc” mode of communication.

When using the 802.11ac infrastructure mode of operation or a similarmode of operations, the AP may transmit a beacon on a fixed channel,such as a primary channel. The primary channel may be a fixed width(e.g., 20 MHz wide bandwidth) or a dynamically set width via signaling.The primary channel may be the operating channel of the BSS and may beused by the STAs to establish a connection with the AP. In certainrepresentative embodiments, Carrier Sense Multiple Access with CollisionAvoidance (CSMA/CA) may be implemented, for example in 802.11 systems.For CSMA/CA, the STAs (e.g., every STA), including the AP, may sense theprimary channel. If the primary channel is sensed/detected and/ordetermined to be busy by a particular STA, the particular STA may backoff. One STA (e.g., only one station) may transmit at any given time ina given BSS.

High Throughput (HT) STAs may use a 40 MHz wide channel forcommunication, for example, via a combination of the primary 20 MHzchannel with an adjacent 20 MHz channel to form a 40 MHz wide contiguouschannel.

Very High Throughput (VHT) STAs may support 20 MHz, 40 MHz, 80 MHz,and/or 160 MHz wide channels. The 40 MHz, and/or 80 MHz, channels may beformed by combining contiguous 20 MHz channels. A 160 MHz channel may beformed by combining 8 contiguous 20 MHz channels, or by combining twonon-contiguous 80 MHz channels, which may be referred to as an 80+80configuration. For the 80+80 configuration, the data, after channelencoding, may be passed through a segment parser that may divide thedata into two streams. Inverse Fast Fourier Transform (IFFT) processing,and time domain processing, may be done on each stream separately. Thestreams may be mapped on to the two 80 MHz channels, and the data may betransmitted by a transmitting STA. At the receiver of the receiving STA,the above described operation for the 80+80 configuration may bereversed, and the combined data may be sent to the Medium Access Control(MAC).

Sub 1 GHz modes of operation are supported by 802.11af and 802.11ah. Thechannel operating bandwidths, and carriers, are reduced in 802.11af and802.11ah relative to those used in 802.11n, and 802.11ac. 802.11afsupports 5 MHz, 10 MHz and 20 MHz bandwidths in the TV White Space(TVWS) spectrum, and 802.11ah supports 1 MHz, 2 MHz, 4 MHz, 8 MHz, and16 MHz bandwidths using non-TVWS spectrum. According to a representativeembodiment, 802.11ah may support Meter Type Control/Machine-TypeCommunications (MTC), such as MTC devices in a macro coverage area. MTCdevices may have certain capabilities, for example, limited capabilitiesincluding support for (e.g., only support for) certain and/or limitedbandwidths. The MTC devices may include a battery with a battery lifeabove a threshold (e.g., to maintain a very long battery life).

WLAN systems, which may support multiple channels, and channelbandwidths, such as 802.11n, 802.11ac, 802.11af, and 802.11ah, include achannel which may be designated as the primary channel. The primarychannel may have a bandwidth equal to the largest common operatingbandwidth supported by all STAs in the BSS. The bandwidth of the primarychannel may be set and/or limited by the particular STA, from among allSTAs operating in a BSS, that supports the smallest bandwidth operatingmode. In the example of 802.11ah, the primary channel may be 1 MHz wideto accommodate STAs (e.g., MTC type devices) that support (e.g., onlysupport) a 1 MHz mode, even if the AP, and other STAs in the BSS support2 MHz, 4 MHz, 8 MHz, 16 MHz, and/or other channel bandwidth operatingmodes. Carrier sensing and/or Network Allocation Vector (NAV) settingsmay depend on the status of the primary channel. If the primary channelis busy, for example, due to a STA (which supports only a 1 MHzoperating mode), transmitting to the AP, the entire available set offrequency bands may be considered busy even though a majority of thefrequency bands remains idle and may be available.

In the United States, the available frequency bands that may be used by802.11ah, are from 902 MHz to 928 MHz. In Korea, the available frequencybands are from 917.5 MHz to 923.5 MHz. In Japan, the available frequencybands are from 916.5 MHz to 927.5 MHz. The total bandwidth available for802.11ah is 6 MHz to 26 MHz depending on the country code.

Spectral efficiency of a WLAN that is an 802.11ac system may be improvedby performing downlink MU-MIMO transmission to more than one STA duringa same symbol time, for example, during a same downlink OFDM symboland/or during a guard interval about the same symbol time. DownlinkMU-MIMO, as implemented by an 802.11ac system, may use the same symboltime to perform downlink transmissions, or in other words,simultaneously transmit symbols, to multiple STAs such that interferenceof a waveform of the downlink transmissions to multiple STAs is not anissue. However, all STAs involved in MU-MIMO transmission with an APmust use a same channel or band, and thus, an operating bandwidth of theMU-MIMO downlink transmissions may be limited to a smallest one of thechannel bandwidths that is supported by the STAs which are included inthe MU-MIMO transmission with the AP.

A WLAN may be an 802.11ad system, wherein the Media Access Control (MAC)layer and Physical (PHY) layer support VHT STAs in the 60 GHz band. An802.11ad system may support data rates up to 7 Gbits/s and may supportthree different modulation modes, including a spread spectrum mode, asingle carrier mode, and an OFDM mode. Furthermore, an 802.11ad systemmay use a 60 GHz unlicensed band, which is available globally. At 60GHz, a wavelength is 5 mm, and an 802.11ad system may have a compactantenna and/or a compact antenna array. Such an antenna and/or antennaarray may transmit and/or receive narrow RF beams, which effectivelyincreases the coverage range and reduces interference in the 802.11adsystem. Additionally, a frame structure for an 802.11ad system allowsfor beamforming training, including discovery and tracking operationsassociated with beamforming. According to one or more embodiments, fourdifferent PHY frame structures is described below, including adescription of sequences, such as a Golay sequence, that may be used inmultiple parts of the different PHY frame structures.

Structures of Four Different PHY Frame Types

FIG. 6 illustrates different PHY frame types according to an embodiment.

Referring to FIG. 6 , a WLAN, such as an 802.11ad system that operatesaccording to an Institute of Electrical and Electronics Engineers (IEEE)802.11ad Directional Multi-Gigabit (DMG) PHY standard, may supportdifferent PHY frame structures, which may also be referred to as frametypes, packets and/or packet structures. According to an embodiment,four different PHY frame types supported by a WLAN, such as an 802.11adsystem, may include a Control PHY frame type 601, a Signal Carrier frametype 602, a Low Power Single Carrier PHY frame type 603, and an OFDM PHYframe type 604.

FIG. 7 illustrates a preamble structure according to an embodiment.

Referring to FIGS. 6 and 7 , the PHY frame types 601 through 604 mayhave a preamble structure that includes a Short Training Field (STF)605/705 and a Channel Estimation Field (CEF) 606/706. Both the STF andthe CEF fields may be constructed from a π/2 (Differential) Binary PhaseShift Keying ((D)BPSK) modulated repeating Golay sequence, as shown inFIG. 7 . According to certain embodiments, the π/2 (D)BPSK modulation ofthe repeating Golay sequence may be performed. However, the presentdisclosure is not limited thereto, and the π/2 (D)BPSK modulation may beperformed according to any suitable variations.

FIG. 8 illustrates a beamforming training field structure according toan embodiment.

Referring to FIG. 8 , a beamforming training field 801 may be includedin a PHY frame at the end of, that is, after or behind, a data field ofthe PHY frame. The beamforming training field 801 may be used for beamrefinement and/or beam training processes. A packet, such as any one ofthe PHY frame types 601 through 604, including the beamforming trainingfield 801 may also be referred to as a Beam Refinement Protocol (BRP)packet.

Referring to FIG. 8 , a training length, which may refer to a length ofthe beam training field 801 may be indicated by N. In some embodiments,as shown in FIG. 8 , an beamforming training field 801 of a PHY framemay include 4N Automatic Gain Control (AGC) subfields 802 and aTraining-Receive/Transmit (TRN-R/T) field 803 includes 5N TRN-R/Tsubfields 804. A Channel Estimation (CE) subfield 805 may be similar toand/or the same as a CEF 606/706 illustrated in FIG. 6 and FIG. 7 .According to other embodiments, N may be the number of antenna weightvectors to be trained, wherein one AGC block may include four subfieldsand one TRN-R/T block may include five subfields, such as 4 R/Tsequences and one CEF, and additionally, all subfields, including theAGC subfield 802 and the TRN-R/T subfield 803 may be modulated usingrotated π/2-BPSK modulation prior to being transmitted.

A Golay sequence, used to construct a CEF and a STF, may be definedaccording to the following. According to an embodiment, ρ_(a)(k) may bethe aperiodic autocorrelation of the sequence a={a₀, a₁, . . . ,a_(N-1)} and ρ_(a)(k) is explicitly given by

${\rho_{a}(k)} = \left\{ \begin{matrix}{{\sum\limits_{i = 0}^{N - k - 1}{a_{i}a_{({i + k})}^{*}}},\ {k \in \left\lbrack {0,\ {N - 1}} \right\rbrack}} \\{{\sum\limits_{i = {|k|}}^{N - 1}{a_{i}a_{({i + k})}^{*}}},\ {k \in \left\lbrack {{- N} + L - 1} \right\rbrack}}\end{matrix} \right.$wherein (⋅)* is the conjugate of its argument. Then, the pair of (a, b)may be referred to as a Golay complementary pair if ρ_(a)(k)+ρ_(b)(k)=0,k≠0.

According to certain embodiments, Golay complementary pairs andsequences may be used for peak-to-average power mitigation, estimationof IQ imbalance parameters, and channel estimation due to its uniquefeatures. Golay sequences may have a property, which may be beneficiallyused in communication systems, wherein out-of-phase periodicautocorrelation of the sequences is zero. That is, multiple delayedreplicas, or in other words, repetitions, of the Golay sequences mayarrive at a receiver due to a multipath in wireless communication. Insome embodiments, in order to estimate a channel, a shifted version ofthe Golay sequence may be orthogonal to another shifted version of thesame sequence. A periodic autocorrelation of a Golay sequence a may begiven as c_(a)(k) as shown below.

${{c_{a}(k)} = {\sum\limits_{i = 0}^{N - k - 1}{a_{i}a_{mod\ {\{{{({i + k})},N}\}}}^{*}}}},{{❘k❘} \in \left\lbrack {0,{N - 1}} \right\rbrack}$where mod{a, N} is modulo N of a. According to some embodiments, theperiodic autocorrelation of a Golay sequence a may also be expressedusing the definition of an aperiodic autocorrelation of the Golaysequence a, as given below.

${c_{a}(k)} = \left\{ \begin{matrix}{{{\rho_{a}(k)} + {\rho_{a}\left( {k - N} \right)}}\ ,} & {k \in \left\lbrack {0,\ {N - 1}} \right\rbrack} \\{{{\rho_{a}(k)} + {\rho_{a}\left( {k + N} \right)}}\ ,} & {k \in \left\lbrack {{{- N} + 1},\ {- 1}} \right\rbrack}\end{matrix} \right.$Therefore, the following expression also holds true for the Golaycomplementary pairs:c _(a)(k)+c _(b)(k)=0,k≠0.This property of Golay complementary pairs, wherein the sum of theperiodic and aperiodic autocorrelations of the Golay sequence a is zero,may be used by the receiver to estimate each channel tap by usingcorrelation operations.

A beam training field of length N=2^(M) Golay complementary pairs may beconstructed according to the recursive procedure show below.a _(k) ^((m)) =w _(m) a _(k) ^((m-1)) +b _(k-d) _(m) ^((m-1))b _(k) ^((m)) =w _(m) a _(k) ^((m-1)) −b _(k-d) _(m) ^((m-1))

where a_(k) ⁽⁰⁾=a_(k) ⁽¹⁾=δ_(k), δ_(k) is the Kronecker's delta, andw_(m) is the m th element of rotation vector w=[w₁ w₂ . . . w_(m)] where|w_(m)|=1, d_(m) is the m th element of the delay vector d=[d₁ d₂ . . .d_(m)] and the permutation of [1 2 . . . 2^(M)].

According to embodiments, in the case of a WLAN that is an 802.11adsystem, the pairs of Golay complementary sequences may be generatedbased on the aforementioned method and three pairs are considered:(Ga₃₂, Gb₃₂), (Ga₆₄, Gb₆₄), and (Ga₁₂₈, Gb₁₂₈). The parameters of thesepairs are listed as follows:

-   -   Ga₃₂=flip{_(k) ⁽⁵⁾} and Gb₃₂=flip{b_(k) ⁽⁵⁾}:    -   w=[−1 1 −1 1 −1] and d=[1 4 8 2 16]    -   Ga₆₄=flip{a_(k) ⁽⁶⁾} and Gb₆₄=flip{b_(k) ⁽⁶⁾}:    -   w=[1 1 −1 −1 1 −1] and d=[2 1 4 8 16 32]    -   Ga₁₂₈=flip{a_(k) ⁽⁷⁾} and Gb₁₂₈=flip{a_(k) ⁽⁷⁾}:    -   w=[−1 −1 −1 −1 1 −1 −1] and d=[1 8 2 4 16 32 64]        Where flip{⋅} is a function that reverses an order of its        argument. According to another embodiments, in the case of the        802.11ad system, the Golay sequences may be used in a SC PHY        frame, for example, at locations indicated by Ga₆₄, and in a low        power SC PHY, for example, at locations indicated by Ga₆₄ and        Gb₁₂₈, as well as in a beamforming training field.

IEEE has approved Task Group ay (TGay) to develop an amendment thatdefines standardized modifications to both the IEEE 802.11 PHY layer andthe IEEE 802.11 MAC layer that enables at least one mode of operationcapable of supporting a maximum throughput of at least 20 Gb/s asmeasured at the MAC data service access point, while maintaining orimproving the power efficiency per station. This amendment also definesoperations for license-exempt bands above 45 GHz while ensuring backwardcompatibility and coexistence with legacy directional multi-gigabitstations, as defined by IEEE 802.11ad-2012 amendment, operating in thesame band.

Although the maximum throughput of at least 20 Gb/s is a primary goal ofTGay, inclusion of mobility and outdoor support has also been proposed.More than ten different use cases are proposed and analyzed in terms ofthroughput, latency, operation environment and applications. Since802.11 ay may operate in the same band as legacy standards, it may berequired that the new technology provide backward compatibility andcoexistence with legacies in the same band.

To reach the maximum throughput requirement for the millimeter wave(mmW) band used in 802.11ad and 802.11 ay, multiple technologies havebeen proposed, including, Single-User-MIMO (SU-MIMO), i.e., multiplestreams, channel bonding, higher order modulation and non-uniformmodulation. To support more features, wider usage scenarios, e.g.,outdoor and mobility use cases, and overall system capacity, othertechnologies may be included in 802.11ay, for example Multi-User-MIMO(MU-MIMO) and enhanced relay.

FIG. 9 illustrates a flowchart of performing MIMO transmissions.

Referring to FIG. 9 , information bits 900, which may also be referredto as data information bits 900, are inputted into an encoder parser 902and one or more of encoders 904 in order to generate encoded data atoperation 901. According to embodiments, a multiple encoder may be usedwhen a packet size is very large. A coding rate of each of the encoders904 may be selected from a modulation and coding scheme (MCS) table.

At operation 903, the encoded data, as generated at operation 901, maybe provided to a spatial parser 906 in order to generate a plurality ofspatial streams (SSs) 908 that are simultaneously transmitted to areceiver using MIMO technology. A quantity Nss may indicate a number ofSSs in the plurality of SSs. At operation 905, respective binarysequences of each of the plurality of SSs 908 are mapped to the complexdomain by a constellation mapper 910. According to an embodiment, theconstellation mapper 910 may map the plurality of SSs 908 based on oneor more of QPSK, 16QAM, π/2-BPSK, and/or any other similar and/orsuitable modulation scheme. In some embodiments, differentconstellations and/or modulation schemes may be applied to differentones of the plurality of SSs 908, and a constellation and/or amodulation scheme to be applied may be determined based on the MCStable. An output of the constellation mapper 910 may be referred to ascomplex modulated symbols, and, according to an embodiment, the complexmodulated symbols may be included in a data field of a packet, such as aPHY packet.

At operation 907, a preamble, which may also be a midamble and/or apostamble, may be attached to the data field of each PHY packetgenerated at operation 905, wherein the preamble includes one or more ofa STF, a CEF, and a header field. According to an embodiment, a same ora different complex modulation applied to the data field may be appliedto the fields included in the preamble. According to some embodimentssupporting MIMO communications, orthogonal CEFs or low cross-correlationCEFs may be used for different spatial streams, such as differentspatial streams among the plurality of SSs 908. At operation 909, aspatial mapper 912 processes the complex modulated symbols of allspatial streams of the plurality of SSs 908. The spatial mapper 912 mayperform one or more of a Space-Time Block Coding (STBC), a CyclicShifting (for cyclic shifting diversity), and spatial multiplex recodingand beamforming. An output of the spatial mapper 912 may be referred toas a transmit (TX) chain.

According to certain embodiments, in a case where the PHY packet is anOFDM PHY packet, the complex symbols in each TX chain outputted by thespatial mapper 912 may be mapped to OFDM symbols and generatedtime-domain signals based on an inverse-discrete Fourier transform(IDFT) operation by a multi-carrier modulator 914, at operation 911.According to an embodiment, a guard interval (GI) may be attached totime-domain samples for each OFDM symbol, and a windowing operation maybe performed after the attaching of the GI. According to an embodiment,the complex symbols in each TX chain may be transmitted from one or moreDMG antenna chains. According to embodiments supporting simultaneous BRPoperations, rather than sequential BRP operations, for each of the oneor more DMG antenna chains, orthogonal beamforming training fields orlow cross-correlation beam training fields, which may be referred to astraining (TRN) fields, may be attached to the data field along with thepreambles and header fields of respective DMG antenna chains, atoperation 913. In a case where the DMG antenna chains for different TXchains use a same set of TRN fields, the beam refinement process fordifferent TX chains may be done sequentially.

According to embodiments, outputs of operation 913, for each of the DMGantenna chains, are provided to respective digital/analog converters 916at operation 915, and analog forms of the DMG antenna chains may bebeamformed by respective analog precoders 918 at operation 917. Theanalog precoders 918 may be referred to as analog beamformers and mayoperate independently and/or collaboratively.

Method of Providing Backward Compatible CEFs for MIMO Transmissions

Multiple spatial layer transmission technology, for example SU-MIMO, maybe used for providing and/or exceeding a 20 Gb/s throughput via multipletransmit antennas. To enable SU-MIMO, CEFs should be mutuallyorthogonal, that is, the CEFs should be generated and/or constructed soas to be mutually orthogonal CEFs, one for each spatial layer. Sincebackward compatibility and coexistence with legacy 802.11ad devices areneeded for 802.11ay, a type of one of the CEFs used for 802.11ay may bethe same as or similar to a type of CEF used in 802.11ad. Accordingly,an 802.11ay system needs to identify one or more CEFs such that the newCEFs and the CEF in 802.11ad are mutually orthogonal to each other andhave the same and/or similar properties.

According to embodiments, two streams may be transmitted at the sametime, or in other words, transmitted simultaneously. In such a case,s_(uv)∈

^(1×1280) and s_(vu)∈

^(1×1280) may be sequences respectively including the last 128 samplesof STFs and CEFs of single carrier and OFDM transmissions in an IEEE802.11ad system, where

is the field of complex numbers. Thus, according to embodiments, thevectors s_(uv) and s_(vu) may be given by:

-   -   s_(uv)=[−m_(Ga) ₁₂₈ m_(Gu) ₅₁₂ m_(Gv) ₅₁₂ −m_(Gb) ₁₂₈ ]    -   s_(vu)=[−m_(Ga) ₁₂₈ m_(Gv) ₅₁₂ m_(Gu) ₅₁₂ −m_(Gb) ₁₂₈ ]        where m_(Ga) ₁₂₈ ∈        ^(1×128), m_(Gu) ₅₁₂ ∈        ^(1×512), m_(Gv) ₅₁₂ ∈        ^(1×512), and m_(Gb) ₁₂₈ ∈        ^(1×128) are modulated Golay sequences, which may be expressed        as:        m _(Ga) ₁₂₈ =Ga ₁₂₈ ⊙r ₁₂₈        m _(Gu) ₅₁₂ =Gu ₅₁₂ ⊙r ₅₁₂        m _(Gv) ₅₁₂ =Gv ₅₁₂ ⊙r ₅₁₂        m _(Gb) ₁₂₈ =Gb ₁₂₈ ⊙r ₅₁₂        where Ga₁₂₈∈        ^(1×128), Gu₅₁₂∈        ^(1×512), Gv₅₁₂∈        ^(1×512), Gb₁₂₈∈        ^(1×128) are Golay vectors that may be defined according to the        IEEE 802.11ad standard,        is the field of real numbers,

$r_{N} = \left\lbrack {e^{j\frac{\pi}{2}0}e^{j\frac{\pi}{2}1}\ \ldots e^{j\frac{\pi}{2}{({N - 1})}}} \right\rbrack$which is for rotating the entries of the Golay vectors in the complexplane, and ⊙ is the Hadamard product.

FIG. 10 illustrates CEFs for multiple streams of a single carriertransmission according to an embodiment.

Referring to FIG. 10 , a first sequence for stream 1 may come fromlegacy STF and legacy CEF(s). The first sequence for stream 1 may comefrom an MS preamble. In some embodiments, the vector s_(uv) maycorrespond to the first sequence including the last part of the STF andthe CEFs for an 802.11ad Single Carrier (SC) transmission, and s_(vu)may correspond to a second sequence for the last part of STFs and theCEFs of an 802.11ad OFDM transmission. In the case of a multipathchannel, the vectors −m_(Ga) ₁₂₈ and −m_(Gb) ₁₂₈ located at the rightand left side of the vectors of s_(uv) and s_(vu) allow for the shiftedversions s_(uv) and s_(vu) yield circular shifts for m_(Gu) ₅₁₂ andm_(Gv) ₅₁₂ .

According to certain embodiments, in a case where two parallel streamsare transmitted, the transmitter may generate the CEFs, including thelast 128 samples of the STFs, of the streams by using the sequencesdefined in the 802.11ad standard, i.e., s_(uv) and s_(vu), and theirconjugates, i.e., s*_(uv), and s*_(vu). In further detail, s_(uv) ands*_(uv), may be used for streams in a SC transmission, and s_(vu) ands*_(vu), may be used for stream in an OFDM transmission. Furthermore, byusing the pair of s_(uv) and s*_(uv), or the pair of s_(vu) and s*_(vu),a maximum separation between the streams may be achieved, wherein themaximum separation between the streams is shown by the followingobservations:

-   -   (a) circshift{m_(Gu) ₅₁₂ , τ}⊥m_(Gu) ₅₁₂ |τ|≤128, τ≠0    -   (b) circshift{m_(Gv) ₅₁₂ , τ}⊥m_(Gv) ₅₁₂ |τ|≤128, τ≠0    -   (c) circshift{m*_(Gu) ₅₁₂ , τ}⊥m*_(Gu) ₅₁₂ |τ|≤128, τ≠0    -   (d) circshift{m*_(Gv) ₅₁₂ , τ}⊥m*_(Gv) ₅₁₂ |τ|≤128, τ≠0    -   (e) circshift{m_(Gu) ₅₁₂ , τ}⊥m*_(Gu) ₅₁₂ |τ|≤128, τ≠0    -   (f) circshift{m_(Gv) ₅₁₂ , τ}⊥m*_(Gv) ₅₁₂ |τ|≤128, τ≠0        wherein the circshift{a, n} is an operator which applies n        circular shifts from the left to right to the sequence a.

In certain embodiments, modulated symbol sequences used for CEFs may bedesigned for two data stream transmission. For example, a CEF for afirst stream may have format [m_(Gu512), m_(Gv512) −m_(Gb128)], and aCEF for a second stream may have format [m_(Gu512)*, m_(Gv512)*−m_(Gb128)*]. The modulated symbols used for the CEFs for the firststream and the second stream are illustrated in Table A. The firstcolumn may show a first modulated symbols corresponding to the CEF forthe first stream with π/2 BPSK modulation. The second column may show asecond modulated symbols corresponding to the CEF for the second streamwith π/2 BPSK modulation. The third column may show a first modulatedsymbols corresponding to the CEF for the first stream without π/2 BPSKmodulation. The fourth column may show a second modulated symbolscorresponding to the CEF for the second stream without π/2 BPSKmodulation. The operator ⊙ corresponds to Hadamard product.

FIG. 11 illustrates autocorrelation zones and cross correlation zones ofsequences s_(uv) and s_(vu) according to an embodiment.

Referring to FIG. 11 , the operations shown in (a)-(d) may indicate thatm_(Gu) ₅₁₂ , m_(Gv) ₅₁₂ and their conjugates have zero autocorrelationzones, the operations shown in (e)-(f) may indicate that the pair ofm_(Gu) ₅₁₂ and m*_(Gu) ₅₁₂ and the pair of m_(Gv) ₅₁₂ and m*_(Gv) ₅₁₂have zero cross-correlation zones. Since the number of elements in thevector m_(Gu) ₅₁₂ is 512, the size of zero autocorrelation zone is 256,and the size of the zero cross-correlation zone is 256. Thus, accordingto embodiments, the proposed sequence may exploit all availabledegrees-of-freedom and may yield the maximum separation between thesequences. According to embodiments, because the sequences s_(uv) ands_(vu) have maximum separation, both 802.11ad and 802.11ay receivers mayestimate the channels with ±128 taps by using Golay correlators.

According to certain embodiments, in a MIMO system and/or with respectto designing a MIMO system, orthogonality of a CEF (for example, a pilotsignal, and/or a reference sequence) for multiple TX antennas may beimplemented, executed, and/or achieved based on techniques similar tospace-time coding. In some embodiments, in a case of a single TXantenna, a transmission time of a CEF may be increased by NT times(wherein, N_(T) is a number of TX antennas). In other embodiments, in acase of more than one TX antenna, for example, N_(T)=2, one CEF (forexample, which may have an increased transmission time) may be usedinstead of using two CEF. According to certain embodiments, and asdiscussed below, increasing the transmission time of CEFs may beextended and/or applied to cases with N_(T)>2, such as a case whereN_(T)=4.

FIG. 12 illustrates extended CEFs for multiple streams according to anembodiment.

Referring to FIG. 12 , a Signal-to-Noise Ratio (SNR) of an estimatedchannel may be increased. According to embodiments, parts of a CEF (forexample, [m_(Gu) ₅₁₂ m_(Gv) ₅₁₂ ] parts of a CEF) may be extended byappending and/or repeating m_(Gu) ₅₁₂ , m*_(Gu) ₅₁₂ , m_(Gv) ₅₁₂ , andm*_(Gv) ₅₁₂ in a CEF. According to certain embodiments, the extendedCEFs may be used in certain cases, however, a CEF(s) that is notextended may be considered to be a default and/or baseline CEFstructure, e.g., as used in many cases. According to certainembodiments, parts of a CEF may be extended by appending a correspondingcyclic suffix (for example, −m_(Gb) ₁₂₈ and/or −m*_(Gb) ₁₂₈ ) in a CEF.FIG. 12 , at parts (a) and (b) illustrates respective extensions ofCEFs. However, the present disclosure is not limited thereto, and areduced CEF may include any suitable and/or similar element forincreasing a SNR.

FIG. 13 illustrates reduced CEFs for multiple streams according to anembodiment.

Referring to FIG. 13 , a latency associated with a channel and/or asignal may be reduced. According to embodiments, a reduced CEF may beused to decrease associated latency. For example, a reduced CEF mayinclude only m_(Gu) ₅₁₂ or m_(Gv) ₅₁₂ . However, the present disclosureis not limited thereto, and a reduced CEF may include any suitableand/or similar element for reducing latency.

Method of Transmitting a Preamble and Header with Multiple TransmitAntennas

In order to provide backward compatibility of 802.11ay systems withlegacy 802.11ad systems, a definition of the transmission method of apreamble, including a STF and a CEF, and the header part of the packetis needed.

FIG. 14 illustrates interoperability of 802.11ad and 802.11ay systemsaccording to an embodiment.

Referring to FIG. 14 , an 802.11ad receiver 1401, which may be an802.11ad STA, an 802.11ay transmitter 1402, which may be an 802.11ay AP,and an 802.11ay receiver 1403, which may be an 802.11ay STA, areillustrated. According to embodiments, in order for the 802.11adreceiver 1401 to successfully decode, or in other words correctly decodea received signal, the cross channel for Stream 1, i.e., h_(11ad), andthe MIMO channels, i.e., h₁₁, h₁₂, h₂₁, and h₂₂, may be estimatedrespectively at the 802.11ad receiver 1401 and at the 802.11 ay receiver1402.

FIG. 15 illustrates channel estimation results of the 802.11ad receiver;and FIG. 16 illustrates channel estimation results of the 802.11ayreceiver according to an embodiment.

Referring to FIGS. 14, 15 and 16 , according to embodiments using theaforementioned zero correlation zone properties of the Golay sequence,the estimated channel results at the 802.11ad receiver 1401 are shown inFIG. 15 , and the estimated channel results at the 802.11ay receiver1403 are shown in FIG. 16 , for a random exponential delay channel.According to embodiments, perfect channel estimations may be achievedfor both cross-channel and MIMO channels.

FIG. 17 illustrates a header for an 802.11ay packet according to anembodiment.

Referring to FIG. 17 , according to an embodiment, in order to transmitinformation related to the streams, extra header fields may be includedfor an embodiment using the aforementioned zero correlation zoneproperties of the Golay sequence, while providing a minimal overheaddesign for SU-MIMO transmissions. As shown in FIG. 17 , headerinformation related to a first stream 1701 and a second stream 1702 aretransmitted over both the first and second streams 1701 and 1702. Inorder to minimize the overhead of the 802.11 ay packet, existing bits ina legacy header 1703 may be used for the first stream 1701. Informationrelated to new features of 802.11ay, e.g., information related toSU-MIMO, may be transmitted over the second stream 1702. According toembodiments, a format of the legacy header 1703 may be kept constant andthe energy level for the second stream 1702 may be kept lower than thatof the first stream 1701 so that the legacy header 1703 is decodable at802.11ad receivers.

Method of Providing a CEF for Beam Tracking for Multiple Streams

According to embodiments, a BRP may be used to provide a needed linkbudget between a pair of STAs. For example, a WLAN that is an 802.11adsystem may use a BRP for receiver training and iterative refinement ofthe antenna settings of both a transmitter and a receiver of both of thepair of STAs. For BRP, the channel estimation field defined for 802.11admay be employed and transmitted repetitively during beam training and/orbeam refinement operations. In a case in which multiple streams aresupported, CEFs which allow the use of existing BRP may be needed.

FIG. 18 illustrates Channel Estimation Fields (CEFs) for two streams,according to an embodiment.

Referring to FIGS. 17 and 18 , a first stream 1801 and a second stream1802 may be use by BRP, or in other words, may be included in a BRPmessage. According to embodiments, a receiver, for example, an 802.11aySTA, an 802.11ay AP, and/or any other suitable and/or similar device,may simultaneously track two beams. In such a case, second CEFs 1804 ofthe second stream 1802, are used for BRP, or in other words, correspondto a second stream included in a BRP message, and a sector level sweep(SLS) message. The second CEFs are conjugates of first CEFs 1803 of thefirst stream 1801 used for BRP and SLS. According to embodiments, and asdescribed above, the use of conjugate CEFs for the second stream 1802may allow maximum separation between the first and second streams 1801and 1802, and may yield perfect and/or approximately perfect channelestimation for antenna ports, for pairs of transmit antenna ports,and/or for an input of digital precoder of a transmitter, and a receiveantenna port and/or an output of a digital precoder of a receiver, whena 2×2 MIMO system is considered. However, the present disclosure is notlimited thereto, and two or more streams may be used for BRP and/orincluded in a BRP message.

In further detail, a CEF, such as the first and/or second CEFs 1603 and1604, may be extended to allow N data stream MIMO transmissions, i.e.,where N≥2 as described below. According to embodiments, CEFs illustratedin FIGS. 23 and 24 may also be used as and/or are similar to a CEF usedin the beam training fields, or in other words, CEFs used for BRP and/orincluded in a BRP message.

CEF Designs for MIMO

According to embodiments, a method of tracking more than one stream maybe applied to N data stream MIMO transmissions, where N≥2, and may beapplied to SU-MIMO and MU-MIMO transmissions.

FIG. 19 illustrates an extra CEF according to an embodiment.

According to embodiments, extra CEFs may be inserted and/or used forMIMO channel estimation when the number of space-time streams, which mayalso be referred to as spatial streams, which may also refer to datastreams, and/or streams, is greater than one. The extra CEFs, which maybe referred to, and illustrated, as CEF1 s, may be constructed based onthe basic CEF sequences defined in 802.11ad. According to otherembodiments, the CEF1 s may be extended by prepending the last 128symbols from a STF, as shown in FIG. 19 . According to certainembodiments, a first CEF1 1901 may be referred to as, and/or determinedaccording to, s_(uv)∈

^(1×1280) and may be used for a SC frame type and/or a PHY frame type. Asecond CEF1 1902 may be referred to as, and/or determined according to,s_(vu)∈

^(1×1280) for an OFDM frame type. According to some embodiments, extraCEFs, which may also be referred to as CEF1 s, may be complex modulatedsymbols. Herein below, the term extra CEFs may refer to fields which maybe composed of variations of a CEF1 described with respect to FIG. 19 .However the present disclosure is not limited thereto, and an extra CEFmay be composed, determined, and/or generated according to variationsnot based on the CEF1 described with respect to FIG. 19 .

According to certain embodiments, including embodiments described and/orillustrated herein, a CEF1 and a CEF may be interchangeable. Forexample, a CEF1 may be replaced by a CEF, or vice versa.

FIG. 20 illustrates CEFs for multiple streams according to anembodiment.

According to embodiments, a CEF may be constructed (e.g., configured,designed, etc.) in a variety of manners, types, and/or based on one ormore parameters. According to certain embodiments, a size of a CEF mayvary, e.g., may be changed. According to certain embodiments, a size ofCEF1 may be changed and/or constructed according to a structure. Forexample, a size of a CEF1 may be changed according a structure (e.g., aproper structure) for certain parameters.

According to certain embodiments, in order to improve a SNR of anestimated channel, the [m_(Gu) ₅₁₂ , m_(Gv) ₅₁₂ ] and/or [m_(Gv) ₅₁₂ ,m_(Gu) ₅₁₂ ] pair in a CEF1) may be duplicated (for example, may beextended), as shown in parts (a) and (b) of FIG. 20 . According to otherembodiments, in order to improve a SNR of an estimated channel, the[m_(Gu) ₅₁₂ , m_(Gv) ₅₁₂ ] and/or [m_(Gv) ₅₁₂ m, m_(Gu) ₅₁₂ ] pair in aCEF1 may be reduced (for example, may be shortened) between −m_(Ga) ₁₂₈and −m_(Gb) ₁₂₈ , as shown in parts (c) to (e) of FIG. 20 . According tocertain embodiments, any combination of elements of the aboveembodiments may be combined for improving a SNR of an estimated channel.

According to certain embodiments, a CEF1 may be configured and/orconstructed in a variety of manners, types, and/or based on one or moreparameters. In such a case, a transmitter may signal information relatedto a configuration and/or construction of a CEF1 is constructed.According to certain embodiments, information related to a configurationand/or construction of a CEF1 may be included in a New Header field,which will be discussed below. In any of the embodiments related to FIG.20 , a CEF1 may be replaced by a CEF.

FIG. 21 illustrates a preamble and PHY header for MIMO transmissions,according to an embodiment, and FIG. 22 illustrates a preamble and PHYheader for MIMO transmissions according to another embodiments.

Referring to FIGS. 21 and 22 , according to one or more embodiments, apreamble and a PHY header may be for MIMO transmissions that includefour data streams. However, the present disclosure is not limitedthereto, and a preamble and PHY header may be for MIMO transmissionsthat include N data streams. According to embodiments, a legacy STF(L-STF), a legacy CEF (L-CEF) and a legacy header (L-Header) field maybe the same as used in 802.11ad. Alternatively, different cyclic shiftsmay be applied to a L-STF, a L-CEF, and a L-Header field for a 2^(nd), a3^(rd), and/or a 4^(th) data stream.

According to embodiments, a new header, which may also be referred to asa multi-stream (MS) header, may be inserted after the L-Header field.The new header may be encoded to be explicitly different from the datafield defined in 802.11ad, such that the receiver may notice, or inother words, determine, that the new header is not part of a datatransmission. According to certain embodiments, the new header, whichmay also be referred to as a new header field and/or an MS header field,may contain a parameter used to signal how many extra CEFs follow thenew header field before a data transmission. According to certainembodiments, the new header may contain information related to aconfiguration and/or construction of a CEF1, for example, informationrelated to any of the different configurations of CEFs illustrated inFIG. 21 .

According to embodiments, extra CEFs, which may also be referred to asCEF1 s, may be inserted after the new header field. The receiver may usethe extra CEFs, together with the legacy CEF, for MIMO channelestimations. The number of extra CEFs used may be based on the number ofspace-time streams. According to other embodiments, if M space-timestreams are transmitted, N extra CEFs may be used, wherein N≥M−1. Forexample, if 2 space-time streams are transmitted, then N may be equal to1, if 4 space-time streams are transmitted, then N may be equal to 3,and if 3 space-time streams are transmitted, N may be 2 or 3. However,the present disclosure is not limited thereto, and N may be any valuethat suitably corresponds to M.

The CEF fields, including an extended L-CEF field which may include thelast 128 symbols from a L-STF field and a L-CEF field, and extra CEFs,may be multiplied by an orthogonal matrix V, with size (N+1)×(N+1) asshown in FIG. 19 . In further detail, according to embodiments, allsymbols in the ith CEF field for the j^(th) space-time stream, whereinthe first CEF field may be the extended L-CEF field described above andthe k^(th) CEF field may be the k−1^(th) CEF1 for k>1, may be multipliedby a constant number V_(i,j). The V matrix may be defined as below:

$V = \begin{pmatrix}V_{1,1} & \ldots & V_{1,{N + 1}} \\ \vdots & \ddots & \vdots \\V_{{N + 1},1} & \ldots & V_{{N + 1},{N + 1}}\end{pmatrix}$

According to embodiments, the first column of the V matrix may be fixedto 1. In such a case, the legacy CEF for different streams may not needto be modified and a legacy device may decode the legacy header.Additionally, the V matrix may be designed as an orthogonal matrix, forexample:

$V = {{\begin{bmatrix}1 & 1 & 1 & 1 \\1 & {- i} & {- 1} & i \\1 & {- 1} & 1 & {- 1} \\1 & i & {- 1} & {- i}\end{bmatrix}{or}{}V} = \begin{bmatrix}1 & {- 1} & {- 1} & 1 \\1 & 1 & {- 1} & {- 1} \\1 & {- 1} & 1 & {- 1} \\1 & 1 & 1 & 1\end{bmatrix}}$However, the present disclosure is not limited thereto, and otherorthogonal matrices may be used as well.

Referring to FIG. 22 , according to an embodiment, complex conjugateoperators may be used to create orthogonality between spatial streams.However, the present disclosure is not limited to the choice of extraCEF fields as shown in FIG. 22 . For example, the rows corresponding tostreams may be interchanged, or in other words, the row for a thirdstream 2203 may be disposed between a first stream 2201 and a secondstream 2202, while a fourth stream 2204 may be disposed adjacent to thesecond stream 1902 with respect to the configuration shown in FIG. 22 .

FIG. 23 illustrates a preamble and PHY header for MIMO transmissionsaccording to other embodiments.

Referring to FIG. 23 , the embodiment shown in FIG. 22 may be consideredas an alternative to the embodiment shown in FIG. 21 , using anorthogonal matrix V. According to embodiments, a first column of theorthogonal matrix V may be not fixed to 1s. Thus, starting from thesecond stream, instead of transmitting a L-STF, followed by a L-CEF, theextended L-CEF field, including the last 128 symbols from the L-STF andthe L-CEF, may be multiplied by a coefficient from the orthogonal Vmatrix. Accordingly, V_(k1)*(extended L-CEF) is transmitted, where k isthe stream index. Similarly, instead transmitting a L-Header directlythrough all the streams, V_(k1)*L-Header is transmitted. Thus theantenna configuration of the L-Header field is consistent with that ofthe L-CEF field, and the legacy device may decode the L-Header.According to embodiments, a similar strategy may be applied to the newheader field too, such that V may be any orthogonal matrix, such as a Pmatrix utilized in 802.11n/ac, for example:

$V = \begin{bmatrix}1 & {- 1} & 1 & 1 \\1 & 1 & {- 1} & 1 \\1 & 1 & 1 & {- 1} \\{- 1} & 1 & 1 & 1\end{bmatrix}$

FIG. 24 illustrates a preamble and PHY header according to yet anotherembodiment.

Referring to FIG. 24 , a legacy CEF may not be combined with extra CEFsfor MIMO channel estimation. Thus, according to embodiments, in a caseof an M stream transmission, N=M new CEF fields may be used.

FIG. 25 illustrates a preamble and PHY header according to anembodiment.

Referring to FIG. 25 , a CEF resulting in less overhead is consideredwith respect to the embodiment shown in FIG. 10 . According to certainembodiments, in a case where four streams may be transmitted, only twoCEFs in MS preamble may be inserted after the new header field. Theorthogonality between a first stream and a second stream may bemaintained by the orthogonal matrix V with size 2×2. The first stream isorthogonal with a third stream due to a property of the CEFs shown inthe embodiment of FIG. 10 . The first and fourth streams are mutualorthogonal since the orthogonal matrix V and complex conjugate operatorsare applied. The first, second, third, and fourth streams may be placedin any order. Furthermore, other streams are mutual orthogonal accordingto the similar rules.

According to the embodiment of FIG. 25 , V may be any 2×2 orthogonalmatrix, for example:

$\begin{matrix}{V = {\begin{pmatrix}V_{1,1} & V_{1,2} \\V_{2,1} & V_{2,2}\end{pmatrix} = \begin{pmatrix}1 & 1 \\1 & {- 1}\end{pmatrix}}} & (1)\end{matrix}$Stream indices (e.g., stream 1-stream 4, in FIG. 25 ) and application ofconjugation and orthogonal matrix operations may be exchanged in anyorder. For example: the first stream is orthogonal with the secondstream due to a property of the CEFs as shown in FIG. 10 . The first andthird streams are mutual orthogonal since the orthogonal matrix V andcomplex conjugate operators are applied. Furthermore, other streams aremutual orthogonal according to the similar rules. The design may beextend to any number of spatial stream case by using different size oforthogonal matrix V.

FIG. 26 illustrates a preamble and PHY header according to anotherembodiment.

Referring to FIG. 26 , overhead due to the extra CEFs may be reducedfurther. According to embodiments, an extended L-CEF field may be usedand combined with the extra CEFs for MIMO channel estimation. A fourstream transmission is shown in FIG. 23 to illustrate the CEF and headerdesign, wherein only one extra CEF may be needed. According to theembodiment of FIG. 23 , the V matrix may be defined as (1) (see above),where the elements in the first column are 1s. Thus the L-STF and theL-CEF field transmitted via a first stream and a second stream may bethe same as used in 802.11ad. The extended L-CEF of a third stream and afourth stream may be modified by performing a complex conjugateoperation. A L-Header of the first stream and the second stream may bethe same as used in 802.11ad, while the third stream and the fourthstream may not transmit the L-Header. The second column of the V matrixand the complex conjugate operator may be applied to the CEF1 as shownin FIG. 26 .

According to the above discussed embodiments including a preamble andPHY header for MIMO transmission, the new header field may be used over,or in other words, use for, all the streams included in the MIMOtransmission. However, the present disclosure is not limited thereto,and the new header field may be stream specific.

Method of Estimating a Channel of Multi-Stream Users in Close Proximity

In a case of a transmission to multiple users, the use of mutuallyorthogonal CEFs for each spatial layer and each user is needed, andthus, methods and procedures to identify and utilize the appropriateorthogonality method are needed.

FIG. 27 illustrates a MU transmission having a large degree ofseparation according to an embodiment.

FIG. 28 illustrates a frame structure for a MU transmission having alarge degree of separation according to an embodiment.

FIG. 29 illustrates a MU transmission having a small degree ofseparation according to an embodiment.

FIG. 30 illustrates a frame structure for a MU transmission having asmall degree of separation according to an embodiment.

Referring to FIGS. 27 and 28 , for a MU transmission, analog, digital orhybrid beamforming may be used to separate the streams meant for eachuser. According to embodiments, an extent to which the streams meant foreach user are separated may depend on a degree of separation of theusers; and the degree of separation may be determined by the spatialseparation of the users and the efficacy of the beamforming scheme.According to embodiments, the CEF used in a MU transmission may beadapted based on the degree of separation of the users, allowing for themost efficient CEF to be sent for the MU transmission scenario.

According to embodiments, in a case where the degree of separation islarge, independent streams, each with a respective set of CEFs, may besent simultaneously to respective users, i.e., respective STAs, and theSTAs may successfully perform channel estimation while legacy STAs maybe able to successfully coexist with the transmissions, i.e., may beable to decode the header and decide that the packet is not for them, asshown in FIGS. 27 and 28 .

Referring to FIGS. 29 and 30 , according to another embodiment, in acase where the degree of separation is small, for example, in a casewhere multiple STAs may reside in the same or adjacent beams, additionalmeasures may be needed to ensure that the CEFs are separable and thedata is decodable. Separable CEFs may be designed using methods andprocedures detailed with respect to the embodiments discussed above, asillustrated in FIGS. 29 and 30 . According to embodiments, the headermay contain information, such as a unique ID corresponding to arespective STA, that maps each user, i.e., STA, to the correct stream.According to embodiments, at each receiver the following signal isreceived:y=h _(desired) s _(desired) +h _(undesired) s _(undesired) +nwherein, h_(desired) and h_(undesired) may be known based on theseparable CEFs, and the desired signal may be decoded at each receiver,especially in a case where the energy of the undesired signal is smallerthan the desired signal due to partial spatial orthogonality based ontransmit beamforming.

In such a case, according to embodiments, the following procedure may beperformed. An AP may perform a Sector Level Sweep and Beam RefinementProtocol to identify the best beam for all STAs. The AP may identifySTA1 and STA2 as residing in adjacent beams. The AP may send multi-usertransmission to STA1 and STA2 using a frame structure illustrated inFIG. 24 . The AP may transmit a legacy STF, LTF and SIG on both streams.In this case, each STA may see, or in other words, may receive and/ordetermine, the information as being transmitted in a large delay spreadchannel. The AP may transmit a common header that signals information ona number of users, specific user information, and a stream that eachuser is associated with. The AP may then transmit data to each user, andmay transmit user specific information to each user.

Method of Providing a Training Field for Simultaneous Beam Training andMIMO Transmission

In 802.11ad, the transmitter may have multiple DMG antennas, each ofwhich could be a phased array, a single element antenna, or a set ofswitched beam antennas covered by a quasi-omni antenna pattern. Since802.11ad supports single stream transmission, there is only one TX/RXchain in each transceiver. Therefore, when there are multiple DMGantennas, the beam training has to be done sequentially for each DMGantenna. In addition, in 802.11ay, multiple spatial layers may betransmitted and received simultaneously. This also provides theopportunity for refinement and/or training of steering vectors formultiple spatial layers at the same time. To support this functionality,the beam training field needs to be redesigned.

FIG. 31 illustrates receive/transmit (R/T) sequences for two steams.

Referring to FIG. 31 , according to an embodiment, new beam trainingfields may allow for simultaneous refinement and/or training of thesteering vectors. According to embodiments that provides simultaneousrefinement and/or training of the steering vectors, the receiver mayestimate the channels between each pair of transmit antenna ports,and/or an input of a digital precoder of a transmitter, and a receiveantenna port, and/or an output of a digital precoder of a receiver,while the transmitter/receiver changes its own steering vectors. Forthis purpose, the sequences with perfect autocorrelation and crosscorrelation properties in the beam training fields are highly desirablefor accurate channel estimation. In order to achieve maximum utilizationof the sequence, the modulated R/T sequences of 802.11ad, which arediscussed above with respect to FIG. 8 , and their complex conjugatesmay be employed in the streams. According to embodiments, aconfiguration that may achieve a maximum utilization of the sequence isillustrated in FIG. 31

Referring to FIG. 31 , according to an embodiment, Let T and R be thesequences for R/T of 802.11ad and express the vector T and vector R as:

-   -   T=R=[s_(cp) s_(main)]        wherein,    -   s_(main)=[−m_(Gb) ₁₂₈ m_(Ga) ₁₂₈ m_(Gb) ₁₂₈ m_(Ga) ₁₂₈ ]        and,    -   s_(cp)=[m_(Ga) ₁₂₈ ].        Then, the following arguments (a) through (c), which allow        channel estimations between each pair of transmit antenna ports,        or an input of a digital precoder of a transmitter, and a        receive antenna port, or an output of a digital precoder of a        receiver, must hold true.    -   (a) circshift{s_(main), τ}⊥s_(main)|τ|≤128, τ≠0    -   (b) circshift{s*_(main), τ}⊥s_(main)|τ|≤128, τ≠0    -   (c) circshift{s_(main), τ}⊥s*_(main)*|τ|≤128        While arguments (a) and (b), according to embodiments, may        ensure that the sequences allow perfect channel estimation for        each individual pair of transmit antenna ports or an input of a        digital precoder of a transmitter, and a receive antenna port,        or an output of a digital precoder of a receiver, argument (c)        may ensure that there is no cross-interference between different        channels at the receiver after correlation.

FIG. 32 illustrates zero autocorrelation zones for arguments (a) and (b)and zero cross correlation zones for argument (c), according to anembodiment.

Referring to FIG. 32 , the results for arguments (a), (b), and (c) forthe proposed sequences, i.e., T and T* (or R and R*), are illustrated.

FIG. 33 illustrates beam training fields using complex conjugates of R/Tsequences according to an embodiment.

FIG. 34 illustrates beam training fields using R/T sequences based on anorthogonal V matrix according to an embodiment.

Referring to FIGS. 33 and 34 , according to an embodiment, the methoddescribed above with reference to FIGS. 31 and 32 may be extended toallow beam training fields for N data stream MIMO transmissions by usinga V matrix, which may be an orthogonal V matrix, which is describedabove. According to embodiments, in a case where multiple antennas areused, an 802.11 ay system may choose to partition the multiple antennasinto groups. Parallel beam training, which may also be referred to assimultaneous beam training, may take place for antennas within eachgroup and the groups of the antennas may be trained sequentially. Insome embodiments, in a case having four antennas A1, A2, A3, A4,antennas A1 and A2 may be in a first group G1, and antennas A3 and A4may be in a second group G2. The signals transmitted from the firstgroup G1 use the beam training filed as shown in FIG. 28 , and the samemethod may be applied to the second group G2. Then, the antennas in thefirst group G1 are trained first, and antennas in the second group G2are trained after. However, the present disclosure is not limitedthereto, and the antennas may be grouped in any suitable manner and maybe trained in any suitable sequence and/or in parallel.

Method for Signaling and/or for Auto-Detecting Beam Training

A method and procedure is for signaling whether a sequential,simultaneous, or combination BRP is used is described below. Accordingto embodiments, in a case where simultaneous beamforming training formultiple streams is provided, detecting and/or determining whether thebeamforming training will be done sequentially or simultaneously, and/oras a combination of sequential and parallel methods is needed.Furthermore, signaling of and/or indicating that one or more sequencessupporting simultaneous beamforming training for multiple streams arebeing used, is also needed. Explicit and implicit methods of making theabove noted determinations and/or indications are discussed belowaccording to one or more embodiments.

According to embodiments, re-interpreting and/or re-using reserved bitsin the legacy header fields of a Control PHY frame, a low-power SC PHYframe, a SC PHY frame, and/or an OFDM PHY frame, may provide the abovenoted determinations and/or indications. According to embodiments, in acase of more than one reserved bits, the bits may be continuous reservedbits, or in other words contiguous and/or adjacent reserved bits, may bediscontinuous reserved bits, or may be a combination of continuous anddiscontinuous reserved bits. According to embodiments, in a case of theControl PHY Header, two reserved bits, starting form bit 22, may beused, such that they may be set to 0 to maintain backward capability,and may indicate, to the receiver, use of legacy sequential beamformingtraining only. When the two reserved bits are set to any of non-zerovalue, e.g., 1, 2 or 3, they may implicitly signal that simultaneousbeamforming training will be done by either a parallel method and/or acombination of sequential and parallel methods. In such a case,signaling the one or more sequences supporting simultaneous beamformingtraining for multiple streams may be implemented by one or anycombination of the following methods. According to embodiments, in afirst method, a first reserved bit is set equal to 1 or 0 in order toindicate whether the complex conjugates of the R/T sequences are used tosupport simultaneous beamforming training. According to otherembodiments, in a second method, a second reserved bit may be set equalto 0 or 1 in order to indicate which V matrix is used to supportsimultaneously beamforming training. For example, 0 and 1 may berespectively pre-defined as the index of V_(2×2) and V_(4×4) matrixes.According to yet other embodiments, in a third method, non-zero valuesfor the two reserved bits may indicate three pre-defined V matrixes ifno complex conjugates of the R/T sequence is allowed.

According to embodiments, in a case of the OFDM PHY header, since anOFDM PHY header also has two reserved bits, starting from bit 46, asimilar method and/or the same method described above for the ControlPHY header may be applied to the OFDM PHY header.

According to embodiments, in a case of a low power (LP) single carrier(SC) PHY header or a SC PHY header, there are four reserved bits,starting from bit 44. In a case where two of the four reserved bits arere-used, in order to maintain consistency with the Control PHY headerand the OFDM PHY header, the same methods proposed for the Control PHYheader may be applied to the LP SC/SC PHY header. In a case where allfour of the four reserved bits reused or re-interpreted, signalingsimultaneously beamforming training may be implemented as follows.According to embodiments, the four reserved bits may be set to 0 tomaintain the backward capability and to indicate that the legacysequential beamforming training is to be performed. According to otherembodiments, the four reserved bits may be set to a non-zero value,e.g., 1, 2, . . . , 15 for the four reserved bits, to implicitly signalthat simultaneous beamforming training will be done by either a parallelmethod and/or a combination of sequential and parallel methods. Then,signaling the one or more sequences supporting simultaneous beamformingtraining for multiple streams may be implemented by one or anycombination of the methods described below.

According to embodiments, in a first method, any of the four reservedbits, for example, a first reserved bit set as 1 or 0, may indicatewhether the complex conjugates of the R/T sequences is used to supportsimultaneous beamforming training. According to other embodiments, in asecond method, any of the remaining reserved bits, other than thereserved bit used to indicate the complex conjugates of the R/Tsequences in the first method, may be set to 1 to indicate thecorresponding V matrix used to support simultaneous beamformingtraining. In other embodiments, any one or more of the second, third,and fourth reserved bits may be respectively pre-defined as the index ofV_(2×2), V_(4×4) and V_(8×8) matrices. According to yet anotherembodiments, in a third method, non-zero values for the four reservedbits may indicate any one or more of fifteen pre-defined V matrices in acase where no complex conjugates of the R/T sequence is allowed, oralternatively, may indicate one or more of fourteen pre-defined Vmatrices while enabling use of the complex conjugates of the R/Tsequences.

According to other embodiments, a new header field may be used toindicate information regarding the beams to be trained, i.e.,sequentially, or in parallel, or any combination of both. The new headermay support the combination of both sequential and parallel methods asfollows: N grouped-beams are sequentially trained, and M streams withineach of the grouped-beams are trained in parallel. In a case where Mindicates one stream, the present embodiment may perform the existing802.11ad sequential beamforming training. In a case where N=1, thepresent embodiment may parallel beamforming training. To support thiskind of combination of sequential and parallel training methods, the newheader may be inserted after the Legacy-header field for M streams,and/or the new header field may contain a parameter used to signal thenew sequences used for M simultaneous beamforming training for each of Ngrouped-beams. The legacy header may be the same as in 802.11ad tosignal and/or indicate that the N grouped-beams are sequentiallytrained.

FIG. 35 illustrates a PHY header for a combination of sequential andparallel beamforming training according to an embodiment.

Referring to FIG. 35 , a PHY header for the combination of sequentialand parallel beamforming training, according to an embodiment, mayinclude a legacy header indicating that N grouped-beams are sequentiallytrained, and may include a new header indicating that, for each of Ngrouped-beams, M=2, such that two streams of each of the N grouped beamsare simultaneously trained.

According to other embodiments, auto detection of Beamforming trainingmethod may implicitly signal if the beamforming training for the streamswill be done sequentially or simultaneously and/or signal a combinationof sequential and simultaneous training. In order to distinguishparallel beamforming training or combination of sequential and parallelbeamforming training from legacy sequential beamforming training,switching between different modulations for the legacy headers and thenew headers may indicate and/or signal use of auto-detection ofbeamforming training methods which implicitly signal the selected methodfor the beamforming training. For example, the auto detection ofbeamforming training methods may signal and/or indicate a rotationand/or shift of a same or different degree of phase of the correspondingmodulations of the new header field in a different PHY header, i.e.,π/2-BPSK, π/2-DBPSK, QPSK-OFDM modulations respectively for a LP-SC/SCheader, a control PHY header, and an OFDM PHY header.

FIG. 36 illustrates auto detection of simultaneous beamforming trainingfor a LP-SC/SC PHY header according to an embodiment.

Referring to FIG. 36 , in a LP-SC frame or a SC PHY frame, the legacyheader is modulated by π/2-BPSK, and the new header is signaled and/orindicated to be shifted and/or rotated by π/4 with respect to π/2-BPSK.The π/4-shifted π/2-BPSK results in a constellation of the new headerbeing rotated by 45° relative to the legacy header in a MS SC PHY frame.Upon the receiver detecting and/or determining that the π/2-BPSK isrotated by π/4, the receiver is implicitly signaled to do simultaneousbeamforming training which may be either a parallel beamforming trainingand/or combination of sequential and parallel beamforming training.

According to other embodiments, the streams may be indexed by AutomaticGain Control (AGC) fields of BRP, and the receiver may identify thestreams to be trained by checking and/or determining the sequence at theAGC fields. In such a case:

-   -   c=[c₁ c₂ c₃ c₄]        wherein,    -   c_(i)∈{1, 2, . . . , 16}.        Then, the sequence with for each stream will be:    -   s_(stream)(c)=[s(c₁) s(c₂) s(c₃) s(c₄)]        wherein,

${s(c)} = \left\{ \begin{matrix}m_{Ga_{64}} & {{{if}\ c} = 1} \\{- m_{Ga_{64}}} & {{{if}\ c} = 2} \\m_{Gb_{64}} & {{{if}\ c} = 3} \\{- m_{Gb_{64}}} & {{{if}\ c} = 4} \\m_{Ga_{64}}^{*} & {{{if}\ c} = 5} \\{- m_{Ga_{64}}^{*}} & {{{if}\ c} = 6} \\m_{Gb_{64}}^{*} & {{{if}\ c} = 7} \\{- m_{Gb_{64}}^{*}} & {{{if}\ c} = 8} \\m_{{flip}{\{{Ga_{64}}\}}} & {{{if}\ c} = 9} \\{- m_{{flip}{\{{Ga_{64}}\}}}} & {{{if}\ c} = {10}} \\m_{{flip}{\{{Gb_{64}}\}}} & {{{if}\ c} = {11}} \\{- m_{{flip}{\{{Gb_{64}}\}}}} & {{{if}\ c} = {12}} \\m_{{flip}{\{{Ga_{64}}\}}}^{*} & {{{if}\ c} = {13}} \\{- m_{{flip}{\{{Ga_{64}}\}}}^{*}} & {{{if}\ c} = {14}} \\m_{{flip}{\{{Gb_{64}}\}}}^{*} & {{{if}\ c} = {15}} \\{- m_{{flip}{\{{Gb_{64}}\}}}^{*}} & {{{if}\ c} = {16}}\end{matrix} \right.$wherein, sign(⋅) is signum function and flip{⋅} reverses the order ofthe sequence on its argument.

FIG. 37 illustrates implicit detection of the beams to be trained byusing different sequences in the AGC fields.

Referring to FIG. 37 , different s(c) may also be considered, and areceiver may check and/or determine the one or more sequences toidentify the one or more beams to be trained. For example, if there aretwo streams with the sequences of s_(stream1)([1 2 2 2]) ands_(stream2)([2 2 2 2]), the corresponding sequences are respectivelyobtained as:

-   -   s_(stream1)([1 2 2 2])=[m_(Ga) ₆₄ m_(Gb) ₆₄ m_(Gb) ₆₄ m_(Gb) ₆₄        ]        and    -   s_(stream2)([2 2 2 2])=[m_(Gb) ₆₄ m_(Gb) ₆₄ m_(Gb) ₆₄ m_(Gb) ₆₄        ],

In such a case, according to an embodiment, the receiver checks and/ordetermines if Ga₆₄ exists. Since Ga₆₄ appears on the first quadrant ofthe s_(stream1), the receiver understands and/or determines that only abeam related to stream 1 is going to be trained. The locations of thesequences are shown in FIG. 34 . According to embodiments, a size of thevector c may change depending on an implementation and the different ofGolay sequences, e.g., Golay sequences with different sizes may beconsidered for s(c). Additionally, the explicit and implicit methods ofauto-detection of beam training methods described above may be usedjointly to simplify the auto-detection process with a small increase insignaling overhead. For example, s(c) may be partitioned into twogroups, and signaling may be used to indicate which group is used, suchthat the receiver may detect the AGC field configuration with a smallersearch space.

FIG. 38 illustrates an exemplary flowchart of transmitting informationfor channel estimation for more than one channel for MIMO communicationsin a wireless network.

Referring to FIG. 38 , it is assumed that there are two streams used byBRP or included in a BRP message. According to embodiments, a STA (e.g.,802.11ay STA), an AP (e.g., 802.11ay AP), and/or any other suitabledevice may simultaneously track two streams. For example, an electronicdevice may be configured to determine a first set of complex numberscomprising first channel estimation signal associated with a firstchannel at operation 3810.

The electronic device may also be configured to determine a second setof complex numbers comprising second channel estimation signalassociated with a second channel at operation 3820. According to certainembodiments, the second set of complex numbers are complex conjugates ofthe first set of complex numbers. According to some embodiments, the useof complex conjugates for the second stream may allow maximum separationbetween the first and second streams. For example, in a case of a 2×2MIMO system, the use of complex conjugates for the second stream mayyield perfect and/or approximately perfect channel estimation forantenna ports, for pairs of transmit antenna ports, and/or for an inputof a digital precoder of a transmitter, and a receive antenna portand/or an output of a digital precoder of a receiver.

At operation 3830, the electronic device may be configured to the firstset of complex numbers and the second set of complex numbers via aphysical layer (PHY) frame. However, the present disclosure is notlimited thereto, and more than two streams may be used for BRP and/orincluded in a BRP message.

FIG. 39 illustrates an exemplary flowchart of performing channelestimation of more than one channel for MIMO communications in awireless network.

According to embodiments, a receiver in a STA (e.g., 802.11ay STA), anAP (e.g., 802.11ay AP), and/or any other suitable device maysimultaneously track two streams. Referring to FIG. 39 , an electronicdevice may be configured to receive a first set of complex numberscomprising a first channel estimation signal and a second set of complexnumbers comprising a second channel estimation signal at operation 3910.

Then, the electronic device may be configured to determine, at operation3920, a first channel based on the first channel estimation signal and asecond channel based on the second channel estimation signal. Accordingto some embodiments, the first set of complex numbers and the second setof complex numbers may be received, e.g., via a physical layer (PHY)frame. According to another embodiments, the second set of complexnumbers may be complex conjugates of the first set of complex numbers.According to embodiments, as described above, the use of complexconjugates for the second stream may allow maximum separation betweenthe first and second streams, and may yield perfect and/or approximatelyperfect channel estimation for antenna ports, for pairs of transmitantenna ports, and/or for an input of digital precoder of a transmitter,and a receive antenna port and/or an output of a digital precoder of areceiver, when a 2×2 MIMO system is considered.

FIG. 40 illustrates another exemplary flowchart of performing transmitbeamforming for MIMO communications in a wireless network. According toembodiments, a STA (e.g., 802.11ay STA), an AP, and/or any othersuitable or similar device may be configured to perform these methods ofbeamforming.

Referring to FIG. 40 , an electronic device may be configured to performa SLS and BRP to identify a beam for any and/or all of one or more STAsat operation 4010.

The electronic device may be configured to identify a first STA and asecond STA disposed in adjacent beams at operation 4020. According toembodiments, a header may contain information, such as a unique IDcorresponding to a respective STA, that maps each user (e.g., STA) tothe correct stream. According to embodiments, at each receiver thefollowing signal may be received:y=h _(desired) s _(desired) +h _(undesired) s _(undesired) +nwherein, h_(desired) and h_(undesired) may be known based on theseparable CEFs, and the desired signal may be decoded at each receiver,especially in a case where the energy of the undesired signal is smallerthan the desired signal due to partial spatial orthogonality based ontransmit beamforming.

At operation 4030, the electronic device may be configured to transmit amulti-user transmission to the first STA and the second STA using a PHYframe including information on a first stream corresponding to a firstchannel and information on a second stream corresponding to a secondchannel.

At operation 4040, the electronic device may be configured tosimultaneously transmit user specific information to the first STA andto the second STA in the PHY frame. In some embodiments, the PHY framemay include a legacy STF, a LTF, and a SIG on both the first stream andthe second stream. According to certain embodiments, the PHY frame mayinclude a common header including information on the one or more STAsand information associating the first stream with the first STA andassociating the second stream with the second STA.

Although the solutions described herein consider 802.11 specificprotocols, it is understood that the solutions described herein are notrestricted to this scenario and are applicable to other wireless systemsas well. According to embodiments, methods, apparatuses, and systems forchannel estimation and simultaneous beamforming training for MIMOcommunications, as described herein, with reference to FIGS. 1-40 , maybe applied to any suitable and/or similar wireless systems,communication systems, and/or radio interfaces.

Throughout the solutions and provided examples, the blank areas in thefigures, e.g., white space of the second stream of FIGS. 14 and 18 , mayimply that there is no restriction for this area and any solution can beemployed therein.

Although features and elements are described above in particularcombinations, one of ordinary skill in the art will appreciate that eachfeature or element can be used alone or in any combination with theother features and elements. In addition, the methods described hereinmay be implemented in a computer program, software, or firmwareincorporated in a computer readable medium for execution by a computeror processor. Examples of non-transitory computer-readable storage mediainclude, but are not limited to, a read only memory (ROM), random accessmemory (RAM), a register, cache memory, semiconductor memory devices,magnetic media such as internal hard disks and removable disks,magneto-optical media, and optical media such as CD-ROM disks, anddigital versatile disks (DVDs). A processor in association with softwaremay be used to implement a radio frequency transceiver for use in aWRTU, UE, terminal, base station, RNC, or any host computer.

Moreover, in the embodiments described above, processing platforms,computing systems, controllers, and other devices containing processorsare noted. These devices may contain at least one Central ProcessingUnit (“CPU”) and memory. In accordance with the practices of personsskilled in the art of computer programming, reference to acts andsymbolic representations of operations or instructions may be performedby the various CPUs and memories. Such acts and operations orinstructions may be referred to as being “executed,” “computer executed”or “CPU executed.”

One of ordinary skill in the art will appreciate that the acts andsymbolically represented operations or instructions include themanipulation of electrical signals by the CPU. An electrical systemrepresents data bits that can cause a resulting transformation orreduction of the electrical signals and the maintenance of data bits atmemory locations in a memory system to thereby reconfigure or otherwisealter the CPU's operation, as well as other processing of signals. Thememory locations where data bits are maintained are physical locationsthat have particular electrical, magnetic, optical, or organicproperties corresponding to or representative of the data bits.

The data bits may also be maintained on a computer readable mediumincluding magnetic disks, optical disks, and any other volatile (e.g.,Random Access Memory (“RAM”)) or non-volatile (“e.g., Read-Only Memory(“ROM”)) mass storage system readable by the CPU. The computer readablemedium may include cooperating or interconnected computer readablemedium, which exist exclusively on the processing system or aredistributed among multiple interconnected processing systems that may belocal or remote to the processing system. It is understood that therepresentative embodiments are not limited to the above-mentionedmemories and that other platforms and memories may support the describedmethods.

No element, act, or instruction used in the description of the presentapplication should be construed as critical or essential to theinvention unless explicitly described as such. In addition, as usedherein, the article “a” is intended to include one or more items. Whereonly one item is intended, the term “one” or similar language is used.Further, the terms “any of” followed by a listing of a plurality ofitems and/or a plurality of categories of items, as used herein, areintended to include “any of,” “any combination of,” “any multiple of,”and/or “any combination of multiples of” the items and/or the categoriesof items, individually or in conjunction with other items and/or othercategories of items. Further, as used herein, the term “set” is intendedto include any number of items, including zero. Further, as used herein,the term “number” is intended to include any number, including zero.

Moreover, the claims should not be read as limited to the describedorder or elements unless stated to that effect. In addition, use of theterm “means” in any claim is intended to invoke 35 U.S.C. § 112, ¶6, andany claim without the word “means” is not so intended.

Suitable processors include, by way of example, a general purposeprocessor, a special purpose processor, a conventional processor, adigital signal processor (DSP), a plurality of microprocessors, one ormore microprocessors in association with a DSP core, a controller, amicrocontroller, Application Specific Integrated Circuits (ASICs),Application Specific Standard Products (ASSPs); Field Programmable GateArrays (FPGAs) circuits, any other type of integrated circuit (IC),and/or a state machine.

A processor in association with software may be used to implement aradio frequency transceiver for use in a wireless transmit receive unit(WRTU), user equipment (UE), terminal, base station, Mobility ManagementEntity (MME) or Evolved Packet Core (EPC), or any host computer. TheWRTU may be used m conjunction with modules, implemented in hardwareand/or software including a Software Defined Radio (SDR), and othercomponents such as a camera, a video camera module, a videophone, aspeakerphone, a vibration device, a speaker, a microphone, a televisiontransceiver, a hands free headset, a keyboard, a Bluetooth® module, afrequency modulated (FM) radio unit, a Near Field Communication (NFC)Module, a liquid crystal display (LCD) display unit, an organiclight-emitting diode (OLED) display unit, a digital music player, amedia player, a video game player module, an Internet browser, and/orany WLAN or Ultra Wide Band (UWB) module.

Although the invention has been described in terms of communicationsystems, it is contemplated that the systems may be implemented insoftware on microprocessors/general purpose computers (not shown). Incertain embodiments, one or more of the functions of the variouscomponents may be implemented in software that controls ageneral-purpose computer.

In addition, although the invention is illustrated and described hereinwith reference to specific embodiments, the invention is not intended tobe limited to the details shown. Rather, various modifications may bemade in the details within the scope and range of equivalents of theclaims and without departing from the invention.

TABLE A Modulated CEF for CEF for a a first stream CEF for a ModulatedCEF for a first stream (with pi/2 BPSK second stream second stream, that(without Modulation) (without is Conjugated CEF Symbol pi/2 BPSK[mGu512, mGv512 − pi/2 BPSK [mGu512*, mGv512* − No. Modulation) mGb128]Modulation) mGb128*]. 1 1  1.0000 + 0.0000i 1  1.0000 + 0.0000i 2 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 3 −1  1.0000 − 0.0000i −1 1.0000 + 0.0000i 4 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 5 −1−1.0000 + 0.0000i −1 −1.0000 − 0.0000i 6 −1 −0.0000 − 1.0000i 1−0.0000 + 1.0000i 7 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 8 −1 0.0000 + 1.0000i 1  0.0000 − 1.0000i 9 −1 −1.0000 + 0.0000i −1 −1.0000− 0.0000i 10 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 11 −1  1.0000 −0.0000i −1  1.0000 + 0.0000i 12 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i13 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 14 −1  0.0000 − 1.0000i 1 0.0000 + 1.0000i 15 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 16 1−0.0000 − 1.0000i −1 −0.0000 + 1.0000i 17 1  1.0000 − 0.0000i 1 1.0000 + 0.0000i 18 1 −0.0000 + 1.0000i −1 −0.0000 − 1.0000i 19 −1 1.0000 − 0.0000i −1  1.0000 + 0.0000i 20 −1  0.0000 + 1.0000i 1  0.0000− 1.0000i 21 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 22 1 −0.0000 +1.0000i −1 −0.0000 − 1.0000i 23 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i24 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 25 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 26 1 −0.0000 + 1.0000i −1 −0.0000 − 1.0000i 27 −1 1.0000 + 0.0000i −1  1.0000 − 0.0000i 28 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 29 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 30 1−0.0000 + 1.0000i −1 −0.0000 − 1.0000i 31 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 32 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 33 −1 −1.0000 +0.0000i −1 −1.0000 − 0.0000i 34 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i35 1 −1.0000 − 0.0000i 1 −1.0000 + 0.0000i 36 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 37 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 38 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 39 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 40 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 41 1  1.0000 −0.0000i 1  1.0000 + 0.0000i 42 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i43 1 −1.0000 − 0.0000i 1 −1.0000 + 0.0000i 44 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 45 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 46 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 47 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 48 −1 −0.0000 + 1.0000i 1 −0.0000 − 1.0000i 49 1  1.0000 −0.0000i 1  1.0000 + 0.0000i 50 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i51 −1  1.0000 + 0.0000i −1  1.0000 − 0.0000i 52 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 53 1  1.0000 + 0.0000i 1  1.0000 − 0.0000i 54 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 55 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 56 1  0.0000 − 1.0000i −1  0.0000 + 1.0000i 57 −1 −1.0000 +0.0000i −1 −1.0000 − 0.0000i 58 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i59 −1  1.0000 + 0.0000i −1  1.0000 − 0.0000i 60 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 61 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 62 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 63 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 64 −1 −0.0000 + 1.0000i 1 −0.0000 − 1.0000i 65 −1 −1.0000 +0.0000i −1 −1.0000 − 0.0000i 66 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i67 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 68 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 69 1  1.0000 + 0.0000i 1  1.0000 − 0.0000i 70 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 71 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 72 1  0.0000 − 1.0000i −1  0.0000 + 1.0000i 73 1  1.0000 −0.0000i 1  1.0000 + 0.0000i 74 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i75 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 76 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 77 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 78 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 79 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 80 −1 −0.0000 + 1.0000i 1 −0.0000 − 1.0000i 81 −1 −1.0000 +0.0000i −1 −1.0000 − 0.0000i 82 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i83 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 84 1  0.0000 − 1.0000i −1 0.0000 + 1.0000i 85 −1 −1.0000 − 0.0000i −1 −1.0000 + 0.0000i 86 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 87 −1  1.0000 − 0.0000i −1 1.0000 + 0.0000i 88 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 89 1 1.0000 − 0.0000i 1  1.0000 + 0.0000i 90 −1  0.0000 − 1.0000i 1 0.0000 + 1.0000i 91 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 92 −1 0.0000 + 1.0000i 1  0.0000 − 1.0000i 93 1  1.0000 − 0.0000i 1  1.0000 +0.0000i 94 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 95 −1  1.0000 +0.0000i −1  1.0000 − 0.0000i 96 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i97 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 98 −1 −0.0000 − 1.0000i 1−0.0000 + 1.0000i 99 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 100 1 0.0000 − 1.0000i −1  0.0000 + 1.0000i 101 1  1.0000 + 0.0000i 1  1.0000− 0.0000i 102 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 103 1 −1.0000 +0.0000i 1 −1.0000 − 0.0000i 104 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i105 1  1.0000 + 0.0000i 1  1.0000 − 0.0000i 106 −1  0.0000 − 1.0000i 1 0.0000 + 1.0000i 107 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 108 −1 0.0000 + 1.0000i 1  0.0000 − 1.0000i 109 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 110 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 111 1−1.0000 − 0.0000i 1 −1.0000 + 0.0000i 112 −1  0.0000 + 1.0000i 1  0.0000− 1.0000i 113 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 114 1  0.0000 +1.0000i −1  0.0000 − 1.0000i 115 −1  1.0000 − 0.0000i −1  1.0000 +0.0000i 116 −1 −0.0000 + 1.0000i 1 −0.0000 − 1.0000i 117 1  1.0000 +0.0000i 1  1.0000 − 0.0000i 118 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i119 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 120 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 121 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 122 1−0.0000 + 1.0000i −1 −0.0000 − 1.0000i 123 −1  1.0000 − 0.0000i −1 1.0000 + 0.0000i 124 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 125 −1−1.0000 + 0.0000i −1 −1.0000 − 0.0000i 126 1  0.0000 + 1.0000i −1 0.0000 − 1.0000i 127 1 −1.0000 − 0.0000i 1 −1.0000 + 0.0000i 128 −1 0.0000 + 1.0000i 1  0.0000 − 1.0000i 129 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 130 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 131 1−1.0000 + 0.0000i 1 −1.0000 − 0.0000i 132 1  0.0000 − 1.0000i −1 0.0000 + 1.0000i 133 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 134 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 135 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 136 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 137 1  1.0000 +0.0000i 1  1.0000 − 0.0000i 138 −1  0.0000 − 1.0000i 1  0.0000 + 1.0000i139 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 140 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 141 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 142 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 143 1 −1.0000 − 0.0000i 1−1.0000 + 0.0000i 144 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 145 −1−1.0000 + 0.0000i −1 −1.0000 − 0.0000i 146 −1 −0.0000 − 1.0000i 1−0.0000 + 1.0000i 147 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 148 1 0.0000 − 1.0000i −1  0.0000 + 1.0000i 149 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 150 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 151 −1 1.0000 − 0.0000i −1  1.0000 + 0.0000i 152 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 153 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 154 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 155 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 156 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 157 1  1.0000 −0.0000i 1  1.0000 + 0.0000i 158 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i159 −1  1.0000 + 0.0000i −1  1.0000 − 0.0000i 160 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 161 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 162 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 163 −1  1.0000 − 0.0000i −1 1.0000 + 0.0000i 164 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 165 −1−1.0000 + 0.0000i −1 −1.0000 − 0.0000i 166 −1 −0.0000 − 1.0000i 1−0.0000 + 1.0000i 167 −1  1.0000 + 0.0000i −1  1.0000 − 0.0000i 168 −1−0.0000 + 1.0000i 1 −0.0000 − 1.0000i 169 −1 −1.0000 − 0.0000i −1−1.0000 + 0.0000i 170 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 171 −1 1.0000 − 0.0000i −1  1.0000 + 0.0000i 172 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 173 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 174 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 175 −1  1.0000 − 0.0000i −1 1.0000 + 0.0000i 176 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 177 −1−1.0000 + 0.0000i −1 −1.0000 − 0.0000i 178 −1  0.0000 − 1.0000i 1 0.0000 + 1.0000i 179 1 −1.0000 − 0.0000i 1 −1.0000 + 0.0000i 180 1 0.0000 − 1.0000i −1  0.0000 + 1.0000i 181 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 182 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 183 −1 1.0000 − 0.0000i −1  1.0000 + 0.0000i 184 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 185 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 186 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 187 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 188 −1 −0.0000 + 1.0000i 1 −0.0000 − 1.0000i 189 1  1.0000 +0.0000i 1  1.0000 − 0.0000i 190 −1  0.0000 − 1.0000i 1  0.0000 + 1.0000i191 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 192 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 193 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 194 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 195 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 196 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 197 1  1.0000 −0.0000i 1  1.0000 + 0.0000i 198 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i199 1 −1.0000 − 0.0000i 1 −1.0000 + 0.0000i 200 1  0.0000 − 1.0000i −1 0.0000 + 1.0000i 201 1  1.0000 + 0.0000i 1  1.0000 − 0.0000i 202 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 203 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 204 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 205 −1 −1.0000 +0.0000i −1 −1.0000 − 0.0000i 206 1  0.0000 + 1.0000i −1  0.0000 −1.0000i 207 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 208 −1  0.0000 +1.0000i 1  0.0000 − 1.0000i 209 −1 −1.0000 − 0.0000i −1 −1.0000 +0.0000i 210 −1  0.0000 − 1.0000i 1  0.0000 + 1.0000i 211 1 −1.0000 −0.0000i 1 −1.0000 + 0.0000i 212 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i213 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 214 −1 −0.0000 − 1.0000i 1−0.0000 + 1.0000i 215 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 216 −1 0.0000 + 1.0000i 1  0.0000 − 1.0000i 217 1  1.0000 − 0.0000i 1 1.0000 + 0.0000i 218 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 219 1−1.0000 + 0.0000i 1 −1.0000 − 0.0000i 220 −1 −0.0000 + 1.0000i 1 −0.0000− 1.0000i 221 1  1.0000 + 0.0000i 1  1.0000 − 0.0000i 222 −1  0.0000 −1.0000i 1  0.0000 + 1.0000i 223 −1  1.0000 − 0.0000i −1  1.0000 +0.0000i 224 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 225 −1 −1.0000 +0.0000i −1 −1.0000 − 0.0000i 226 −1 −0.0000 − 1.0000i 1 −0.0000 +1.0000i 227 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 228 1 −0.0000 −1.0000i −1 −0.0000 + 1.0000i 229 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i230 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 231 1 −1.0000 − 0.0000i 1−1.0000 + 0.0000i 232 1  0.0000 − 1.0000i −1  0.0000 + 1.0000i 233 1 1.0000 + 0.0000i 1  1.0000 − 0.0000i 234 −1 −0.0000 − 1.0000i 1−0.0000 + 1.0000i 235 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 236 −1 0.0000 + 1.0000i 1  0.0000 − 1.0000i 237 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 238 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 239 1−1.0000 + 0.0000i 1 −1.0000 − 0.0000i 240 −1  0.0000 + 1.0000i 1  0.0000− 1.0000i 241 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 242 1 −0.0000 +1.0000i −1 −0.0000 − 1.0000i 243 −1  1.0000 + 0.0000i −1  1.0000 −0.0000i 244 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 245 1  1.0000 −0.0000i 1  1.0000 + 0.0000i 246 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i247 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 248 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 249 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 250 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 251 −1  1.0000 − 0.0000i −1 1.0000 + 0.0000i 252 1  0.0000 − 1.0000i −1  0.0000 + 1.0000i 253 −1−1.0000 − 0.0000i −1 −1.0000 + 0.0000i 254 1 −0.0000 + 1.0000i −1−0.0000 − 1.0000i 255 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 256 −1 0.0000 + 1.0000i 1  0.0000 − 1.0000i 257 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 258 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 259 1−1.0000 + 0.0000i 1 −1.0000 − 0.0000i 260 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 261 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 262 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 263 1 −1.0000 − 0.0000i 1−1.0000 + 0.0000i 264 1  0.0000 − 1.0000i −1  0.0000 + 1.0000i 265 1 1.0000 − 0.0000i 1  1.0000 + 0.0000i 266 −1 −0.0000 − 1.0000i 1−0.0000 + 1.0000i 267 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 268 −1 0.0000 + 1.0000i 1  0.0000 − 1.0000i 269 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 270 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 271 1−1.0000 + 0.0000i 1 −1.0000 − 0.0000i 272 −1  0.0000 + 1.0000i 1  0.0000− 1.0000i 273 −1 −1.0000 − 0.0000i −1 −1.0000 + 0.0000i 274 −1  0.0000 −1.0000i 1  0.0000 + 1.0000i 275 1 −1.0000 − 0.0000i 1 −1.0000 + 0.0000i276 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 277 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 278 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 279 −1 1.0000 − 0.0000i −1  1.0000 + 0.0000i 280 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 281 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 282 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 283 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 284 −1 −0.0000 + 1.0000i 1 −0.0000 − 1.0000i 285 1  1.0000 +0.0000i 1  1.0000 − 0.0000i 286 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i287 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 288 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 289 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 290 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 291 −1  1.0000 − 0.0000i −1 1.0000 + 0.0000i 292 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 293 −1−1.0000 + 0.0000i −1 −1.0000 − 0.0000i 294 −1 −0.0000 − 1.0000i 1−0.0000 + 1.0000i 295 −1  1.0000 + 0.0000i −1  1.0000 − 0.0000i 296 −1−0.0000 + 1.0000i 1 −0.0000 − 1.0000i 297 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 298 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 299 −1 1.0000 − 0.0000i −1  1.0000 + 0.0000i 300 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 301 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 302 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 303 −1  1.0000 − 0.0000i −1 1.0000 + 0.0000i 304 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 305 −1−1.0000 + 0.0000i −1 −1.0000 − 0.0000i 306 −1  0.0000 − 1.0000i 1 0.0000 + 1.0000i 307 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 308 1−0.0000 − 1.0000i −1 −0.0000 + 1.0000i 309 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 310 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 311 −1 1.0000 − 0.0000i −1  1.0000 + 0.0000i 312 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 313 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 314 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 315 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 316 −1 −0.0000 + 1.0000i 1 −0.0000 − 1.0000i 317 1  1.0000 +0.0000i 1  1.0000 − 0.0000i 318 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i319 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 320 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 321 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 322 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 323 −1  1.0000 − 0.0000i −1 1.0000 + 0.0000i 324 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 325 −1−1.0000 + 0.0000i −1 −1.0000 − 0.0000i 326 −1 −0.0000 − 1.0000i 1−0.0000 + 1.0000i 327 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 328 −1 0.0000 + 1.0000i 1  0.0000 − 1.0000i 329 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 330 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 331 −1 1.0000 − 0.0000i −1  1.0000 + 0.0000i 332 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 333 1  1.0000 + 0.0000i 1  1.0000 − 0.0000i 334 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 335 −1  1.0000 + 0.0000i −1 1.0000 − 0.0000i 336 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 337 1 1.0000 + 0.0000i 1  1.0000 − 0.0000i 338 1  0.0000 + 1.0000i −1  0.0000− 1.0000i 339 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 340 −1  0.0000 +1.0000i 1  0.0000 − 1.0000i 341 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i342 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 343 1 −1.0000 + 0.0000i 1−1.0000 − 0.0000i 344 1  0.0000 − 1.0000i −1  0.0000 + 1.0000i 345 −1−1.0000 + 0.0000i −1 −1.0000 − 0.0000i 346 1 −0.0000 + 1.0000i −1−0.0000 − 1.0000i 347 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 348 1 0.0000 − 1.0000i −1  0.0000 + 1.0000i 349 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 350 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 351 1−1.0000 + 0.0000i 1 −1.0000 − 0.0000i 352 −1  0.0000 + 1.0000i 1  0.0000− 1.0000i 353 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 354 1  0.0000 +1.0000i −1  0.0000 − 1.0000i 355 −1  1.0000 + 0.0000i −1  1.0000 −0.0000i 356 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 357 −1 −1.0000 −0.0000i −1 −1.0000 + 0.0000i 358 −1 −0.0000 − 1.0000i 1 −0.0000 +1.0000i 359 −1  1.0000 + 0.0000i −1  1.0000 − 0.0000i 360 −1  0.0000 +1.0000i 1  0.0000 − 1.0000i 361 −1 −1.0000 + 0.0000i −1 −1.0000 −0.0000i 362 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 363 −1  1.0000 −0.0000i −1  1.0000 + 0.0000i 364 1  0.0000 − 1.0000i −1  0.0000 +1.0000i 365 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 366 −1  0.0000 −1.0000i 1  0.0000 + 1.0000i 367 −1  1.0000 − 0.0000i −1  1.0000 +0.0000i 368 1  0.0000 − 1.0000i −1  0.0000 + 1.0000i 369 −1 −1.0000 +0.0000i −1 −1.0000 − 0.0000i 370 −1  0.0000 − 1.0000i 1  0.0000 +1.0000i 371 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 372 1 −0.0000 −1.0000i −1 −0.0000 + 1.0000i 373 −1 −1.0000 + 0.0000i −1 −1.0000 −0.0000i 374 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 375 −1  1.0000 +0.0000i −1  1.0000 − 0.0000i 376 −1  0.0000 + 1.0000i 1  0.0000 −1.0000i 377 1  1.0000 + 0.0000i 1  1.0000 − 0.0000i 378 −1 −0.0000 −1.0000i 1 −0.0000 + 1.0000i 379 1 −1.0000 − 0.0000i 1 −1.0000 + 0.0000i380 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 381 1  1.0000 − 0.0000i 1 1.0000 + 0.0000i 382 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 383 −1 1.0000 − 0.0000i −1  1.0000 + 0.0000i 384 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 385 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 386 −1 0.0000 − 1.0000i 1  0.0000 + 1.0000i 387 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 388 1  0.0000 − 1.0000i −1  0.0000 + 1.0000i 389 1  1.0000 −0.0000i 1  1.0000 + 0.0000i 390 1 −0.0000 + 1.0000i −1 −0.0000 − 1.0000i391 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 392 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 393 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 394 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 395 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 396 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 397 −1 −1.0000 −0.0000i −1 −1.0000 + 0.0000i 398 1  0.0000 + 1.0000i −1  0.0000 −1.0000i 399 1 −1.0000 − 0.0000i 1 −1.0000 + 0.0000i 400 −1  0.0000 +1.0000i 1  0.0000 − 1.0000i 401 −1 −1.0000 − 0.0000i −1 −1.0000 +0.0000i 402 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 403 1 −1.0000 +0.0000i 1 −1.0000 − 0.0000i 404 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i405 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 406 −1 −0.0000 − 1.0000i 1−0.0000 + 1.0000i 407 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 408 −1−0.0000 + 1.0000i 1 −0.0000 − 1.0000i 409 1  1.0000 − 0.0000i 1 1.0000 + 0.0000i 410 −1  0.0000 − 1.0000i 1  0.0000 + 1.0000i 411 1−1.0000 + 0.0000i 1 −1.0000 − 0.0000i 412 −1 −0.0000 + 1.0000i 1 −0.0000− 1.0000i 413 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 414 −1 −0.0000 −1.0000i 1 −0.0000 + 1.0000i 415 −1  1.0000 − 0.0000i −1  1.0000 +0.0000i 416 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 417 1  1.0000 +0.0000i 1  1.0000 − 0.0000i 418 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i419 −1  1.0000 + 0.0000i −1  1.0000 − 0.0000i 420 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 421 −1 −1.0000 − 0.0000i −1 −1.0000 + 0.0000i 422 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 423 −1  1.0000 − 0.0000i −1 1.0000 + 0.0000i 424 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 425 −1−1.0000 + 0.0000i −1 −1.0000 − 0.0000i 426 1  0.0000 + 1.0000i −1 0.0000 − 1.0000i 427 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 428 1 0.0000 − 1.0000i −1  0.0000 + 1.0000i 429 1  1.0000 − 0.0000i 1 1.0000 + 0.0000i 430 −1  0.0000 − 1.0000i 1  0.0000 + 1.0000i 431 −1 1.0000 − 0.0000i −1  1.0000 + 0.0000i 432 1  0.0000 − 1.0000i −1 0.0000 + 1.0000i 433 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 434 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 435 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 436 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 437 −1 −1.0000 +0.0000i −1 −1.0000 − 0.0000i 438 −1 −0.0000 − 1.0000i 1 −0.0000 +1.0000i 439 −1  1.0000 + 0.0000i −1  1.0000 − 0.0000i 440 −1  0.0000 +1.0000i 1  0.0000 − 1.0000i 441 1  1.0000 + 0.0000i 1  1.0000 − 0.0000i442 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 443 1 −1.0000 − 0.0000i 1−1.0000 + 0.0000i 444 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 445 1 1.0000 − 0.0000i 1  1.0000 + 0.0000i 446 −1 −0.0000 − 1.0000i 1−0.0000 + 1.0000i 447 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 448 1−0.0000 − 1.0000i −1 −0.0000 + 1.0000i 449 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 450 −1  0.0000 − 1.0000i 1  0.0000 + 1.0000i 451 1−1.0000 + 0.0000i 1 −1.0000 − 0.0000i 452 1  0.0000 − 1.0000i −1 0.0000 + 1.0000i 453 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 454 1−0.0000 + 1.0000i −1 −0.0000 − 1.0000i 455 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 456 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 457 1  1.0000 −0.0000i 1  1.0000 + 0.0000i 458 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i459 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 460 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 461 −1 −1.0000 − 0.0000i −1 −1.0000 + 0.0000i 462 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 463 1 −1.0000 − 0.0000i 1−1.0000 + 0.0000i 464 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 465 −1−1.0000 − 0.0000i −1 −1.0000 + 0.0000i 466 −1 −0.0000 − 1.0000i 1−0.0000 + 1.0000i 467 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 468 1−0.0000 − 1.0000i −1 −0.0000 + 1.0000i 469 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 470 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 471 −1 1.0000 − 0.0000i −1  1.0000 + 0.0000i 472 −1 −0.0000 + 1.0000i 1−0.0000 − 1.0000i 473 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 474 −1 0.0000 − 1.0000i 1  0.0000 + 1.0000i 475 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 476 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 477 1  1.0000 −0.0000i 1  1.0000 + 0.0000i 478 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i479 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 480 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 481 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 482 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 483 1 −1.0000 − 0.0000i 1−1.0000 + 0.0000i 484 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 485 1 1.0000 + 0.0000i 1  1.0000 − 0.0000i 486 1  0.0000 + 1.0000i −1  0.0000− 1.0000i 487 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 488 1 −0.0000 −1.0000i −1 −0.0000 + 1.0000i 489 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i490 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 491 1 −1.0000 + 0.0000i 1−1.0000 − 0.0000i 492 −1 −0.0000 + 1.0000i 1 −0.0000 − 1.0000i 493 −1−1.0000 + 0.0000i −1 −1.0000 − 0.0000i 494 1 −0.0000 + 1.0000i −1−0.0000 − 1.0000i 495 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 496 −1−0.0000 + 1.0000i 1 −0.0000 − 1.0000i 497 1  1.0000 − 0.0000i 1 1.0000 + 0.0000i 498 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 499 −1 1.0000 − 0.0000i −1  1.0000 + 0.0000i 500 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 501 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 502 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 503 1 −1.0000 − 0.0000i 1−1.0000 + 0.0000i 504 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 505 −1−1.0000 − 0.0000i −1 −1.0000 + 0.0000i 506 1  0.0000 + 1.0000i −1 0.0000 − 1.0000i 507 −1  1.0000 + 0.0000i −1  1.0000 − 0.0000i 508 1−0.0000 − 1.0000i −1 −0.0000 + 1.0000i 509 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 510 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 511 1−1.0000 + 0.0000i 1 −1.0000 − 0.0000i 512 −1  0.0000 + 1.0000i 1  0.0000− 1.0000i 513 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 514 1 −0.0000 +1.0000i −1 −0.0000 − 1.0000i 515 −1  1.0000 − 0.0000i −1  1.0000 +0.0000i 516 −1 −0.0000 + 1.0000i 1 −0.0000 − 1.0000i 517 −1 −1.0000 +0.0000i −1 −1.0000 − 0.0000i 518 −1 −0.0000 − 1.0000i 1 −0.0000 +1.0000i 519 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 520 −1  0.0000 +1.0000i 1  0.0000 − 1.0000i 521 −1 −1.0000 + 0.0000i −1 −1.0000 −0.0000i 522 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 523 −1  1.0000 −0.0000i −1  1.0000 + 0.0000i 524 1 −0.0000 − 1.0000i −1 −0.0000 +1.0000i 525 1  1.0000 + 0.0000i 1  1.0000 − 0.0000i 526 −1 −0.0000 −1.0000i 1 −0.0000 + 1.0000i 527 −1  1.0000 + 0.0000i −1  1.0000 −0.0000i 528 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 529 1  1.0000 −0.0000i 1  1.0000 + 0.0000i 530 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i531 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 532 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 533 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 534 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 535 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 536 1  0.0000 − 1.0000i −1  0.0000 + 1.0000i 537 −1 −1.0000 +0.0000i −1 −1.0000 − 0.0000i 538 1 −0.0000 + 1.0000i −1 −0.0000 −1.0000i 539 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 540 1 −0.0000 −1.0000i −1 −0.0000 + 1.0000i 541 −1 −1.0000 + 0.0000i −1 −1.0000 −0.0000i 542 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 543 1 −1.0000 +0.0000i 1 −1.0000 − 0.0000i 544 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i545 −1 −1.0000 − 0.0000i −1 −1.0000 + 0.0000i 546 −1 −0.0000 − 1.0000i 1−0.0000 + 1.0000i 547 1 −1.0000 − 0.0000i 1 −1.0000 + 0.0000i 548 1−0.0000 − 1.0000i −1 −0.0000 + 1.0000i 549 1  1.0000 + 0.0000i 1  1.0000− 0.0000i 550 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 551 1 −1.0000 +0.0000i 1 −1.0000 − 0.0000i 552 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i553 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 554 −1 −0.0000 − 1.0000i 1−0.0000 + 1.0000i 555 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 556 −1−0.0000 + 1.0000i 1 −0.0000 − 1.0000i 557 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 558 1 −0.0000 + 1.0000i −1 −0.0000 − 1.0000i 559 1−1.0000 + 0.0000i 1 −1.0000 − 0.0000i 560 −1  0.0000 + 1.0000i 1  0.0000− 1.0000i 561 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 562 1  0.0000 +1.0000i −1  0.0000 − 1.0000i 563 −1  1.0000 − 0.0000i −1  1.0000 +0.0000i 564 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 565 1  1.0000 −0.0000i 1  1.0000 + 0.0000i 566 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i567 1 −1.0000 − 0.0000i 1 −1.0000 + 0.0000i 568 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 569 −1 −1.0000 − 0.0000i −1 −1.0000 + 0.0000i 570 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 571 −1  1.0000 − 0.0000i −1 1.0000 + 0.0000i 572 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 573 −1−1.0000 + 0.0000i −1 −1.0000 − 0.0000i 574 1  0.0000 + 1.0000i −1 0.0000 − 1.0000i 575 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 576 −1 0.0000 + 1.0000i 1  0.0000 − 1.0000i 577 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 578 −1  0.0000 − 1.0000i 1  0.0000 + 1.0000i 579 1−1.0000 + 0.0000i 1 −1.0000 − 0.0000i 580 1  0.0000 − 1.0000i −1 0.0000 + 1.0000i 581 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 582 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 583 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 584 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 585 1  1.0000 −0.0000i 1  1.0000 + 0.0000i 586 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i587 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 588 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 589 −1 −1.0000 − 0.0000i −1 −1.0000 + 0.0000i 590 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 591 1 −1.0000 − 0.0000i 1−1.0000 + 0.0000i 592 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 593 −1−1.0000 + 0.0000i −1 −1.0000 − 0.0000i 594 −1 −0.0000 − 1.0000i 1−0.0000 + 1.0000i 595 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 596 1−0.0000 − 1.0000i −1 −0.0000 + 1.0000i 597 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 598 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 599 −1 1.0000 − 0.0000i −1  1.0000 + 0.0000i 600 −1 −0.0000 + 1.0000i 1−0.0000 − 1.0000i 601 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 602 −1 0.0000 − 1.0000i 1  0.0000 + 1.0000i 603 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 604 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 605 1  1.0000 −0.0000i 1  1.0000 + 0.0000i 606 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i607 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 608 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 609 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 610 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 611 1 −1.0000 − 0.0000i 1−1.0000 + 0.0000i 612 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 613 1 1.0000 − 0.0000i 1  1.0000 + 0.0000i 614 1  0.0000 + 1.0000i −1  0.0000− 1.0000i 615 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 616 1 −0.0000 −1.0000i −1 −0.0000 + 1.0000i 617 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i618 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 619 1 −1.0000 + 0.0000i 1−1.0000 − 0.0000i 620 −1 −0.0000 + 1.0000i 1 −0.0000 − 1.0000i 621 −1−1.0000 + 0.0000i −1 −1.0000 − 0.0000i 622 1 −0.0000 + 1.0000i −1−0.0000 − 1.0000i 623 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 624 −1 0.0000 + 1.0000i 1  0.0000 − 1.0000i 625 1  1.0000 − 0.0000i 1 1.0000 + 0.0000i 626 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 627 −1 1.0000 − 0.0000i −1  1.0000 + 0.0000i 628 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 629 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 630 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 631 1 −1.0000 − 0.0000i 1−1.0000 + 0.0000i 632 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 633 −1−1.0000 − 0.0000i −1 −1.0000 + 0.0000i 634 1  0.0000 + 1.0000i −1 0.0000 − 1.0000i 635 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 636 1−0.0000 − 1.0000i −1 −0.0000 + 1.0000i 637 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 638 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 639 1−1.0000 + 0.0000i 1 −1.0000 − 0.0000i 640 −1  0.0000 + 1.0000i 1  0.0000− 1.0000i 641 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 642 1 −0.0000 +1.0000i −1 −0.0000 − 1.0000i 643 −1  1.0000 − 0.0000i −1  1.0000 +0.0000i 644 −1 −0.0000 + 1.0000i 1 −0.0000 − 1.0000i 645 −1 −1.0000 +0.0000i −1 −1.0000 − 0.0000i 646 −1 −0.0000 − 1.0000i 1 −0.0000 +1.0000i 647 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 648 −1  0.0000 +1.0000i 1  0.0000 − 1.0000i 649 −1 −1.0000 + 0.0000i −1 −1.0000 −0.0000i 650 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 651 −1  1.0000 −0.0000i −1  1.0000 + 0.0000i 652 1 −0.0000 − 1.0000i −1 −0.0000 +1.0000i 653 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 654 −1  0.0000 −1.0000i 1  0.0000 + 1.0000i 655 −1  1.0000 − 0.0000i −1  1.0000 +0.0000i 656 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 657 1  1.0000 −0.0000i 1  1.0000 + 0.0000i 658 1 −0.0000 + 1.0000i −1 −0.0000 − 1.0000i659 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 660 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 661 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 662 1−0.0000 + 1.0000i −1 −0.0000 − 1.0000i 663 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 664 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 665 −1 −1.0000 −0.0000i −1 −1.0000 + 0.0000i 666 1  0.0000 + 1.0000i −1  0.0000 −1.0000i 667 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 668 1 −0.0000 −1.0000i −1 −0.0000 + 1.0000i 669 −1 −1.0000 − 0.0000i −1 −1.0000 +0.0000i 670 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 671 1 −1.0000 +0.0000i 1 −1.0000 − 0.0000i 672 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i673 −1 −1.0000 − 0.0000i −1 −1.0000 + 0.0000i 674 −1 −0.0000 − 1.0000i 1−0.0000 + 1.0000i 675 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 676 1 0.0000 − 1.0000i −1  0.0000 + 1.0000i 677 1  1.0000 − 0.0000i 1 1.0000 + 0.0000i 678 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 679 1−1.0000 + 0.0000i 1 −1.0000 − 0.0000i 680 1  0.0000 − 1.0000i −1 0.0000 + 1.0000i 681 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 682 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 683 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 684 −1 −0.0000 + 1.0000i 1 −0.0000 − 1.0000i 685 −1 −1.0000 +0.0000i −1 −1.0000 − 0.0000i 686 1  0.0000 + 1.0000i −1  0.0000 −1.0000i 687 1 −1.0000 − 0.0000i 1 −1.0000 + 0.0000i 688 −1  0.0000 +1.0000i 1  0.0000 − 1.0000i 689 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i690 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 691 −1  1.0000 + 0.0000i −1 1.0000 − 0.0000i 692 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 693 1 1.0000 − 0.0000i 1  1.0000 + 0.0000i 694 1  0.0000 + 1.0000i −1  0.0000− 1.0000i 695 1 −1.0000 − 0.0000i 1 −1.0000 + 0.0000i 696 1 −0.0000 −1.0000i −1 −0.0000 + 1.0000i 697 −1 −1.0000 + 0.0000i −1 −1.0000 −0.0000i 698 1 −0.0000 + 1.0000i −1 −0.0000 − 1.0000i 699 −1  1.0000 −0.0000i −1  1.0000 + 0.0000i 700 1 −0.0000 − 1.0000i −1 −0.0000 +1.0000i 701 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 702 1 −0.0000 +1.0000i −1 −0.0000 − 1.0000i 703 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i704 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 705 1  1.0000 − 0.0000i 1 1.0000 + 0.0000i 706 1 −0.0000 + 1.0000i −1 −0.0000 − 1.0000i 707 −1 1.0000 − 0.0000i −1  1.0000 + 0.0000i 708 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 709 −1 −1.0000 − 0.0000i −1 −1.0000 + 0.0000i 710 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 711 −1  1.0000 − 0.0000i −1 1.0000 + 0.0000i 712 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 713 −1−1.0000 − 0.0000i −1 −1.0000 + 0.0000i 714 1  0.0000 + 1.0000i −1 0.0000 − 1.0000i 715 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 716 1 0.0000 − 1.0000i −1  0.0000 + 1.0000i 717 1  1.0000 + 0.0000i 1  1.0000− 0.0000i 718 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 719 −1  1.0000 −0.0000i −1  1.0000 + 0.0000i 720 1  0.0000 − 1.0000i −1  0.0000 +1.0000i 721 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 722 1  0.0000 +1.0000i −1  0.0000 − 1.0000i 723 −1  1.0000 − 0.0000i −1  1.0000 +0.0000i 724 −1 −0.0000 + 1.0000i 1 −0.0000 − 1.0000i 725 1  1.0000 −0.0000i 1  1.0000 + 0.0000i 726 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i727 1 −1.0000 − 0.0000i 1 −1.0000 + 0.0000i 728 1  0.0000 − 1.0000i −1 0.0000 + 1.0000i 729 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 730 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 731 −1  1.0000 + 0.0000i −1 1.0000 − 0.0000i 732 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 733 −1−1.0000 + 0.0000i −1 −1.0000 − 0.0000i 734 1  0.0000 + 1.0000i −1 0.0000 − 1.0000i 735 1 −1.0000 − 0.0000i 1 −1.0000 + 0.0000i 736 −1 0.0000 + 1.0000i 1  0.0000 − 1.0000i 737 1  1.0000 − 0.0000i 1 1.0000 + 0.0000i 738 1 −0.0000 + 1.0000i −1 −0.0000 − 1.0000i 739 −1 1.0000 + 0.0000i −1  1.0000 − 0.0000i 740 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 741 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 742 −1 0.0000 − 1.0000i 1  0.0000 + 1.0000i 743 −1  1.0000 − 0.0000i −1 1.0000 + 0.0000i 744 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 745 −1−1.0000 + 0.0000i −1 −1.0000 − 0.0000i 746 1 −0.0000 + 1.0000i −1−0.0000 − 1.0000i 747 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 748 1−0.0000 − 1.0000i −1 −0.0000 + 1.0000i 749 1  1.0000 + 0.0000i 1  1.0000− 0.0000i 750 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 751 −1  1.0000 −0.0000i −1  1.0000 + 0.0000i 752 1 −0.0000 − 1.0000i −1 −0.0000 +1.0000i 753 −1 −1.0000 − 0.0000i −1 −1.0000 + 0.0000i 754 −1 −0.0000 −1.0000i 1 −0.0000 + 1.0000i 755 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i756 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 757 −1 −1.0000 − 0.0000i −1−1.0000 + 0.0000i 758 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 759 −1 1.0000 − 0.0000i −1  1.0000 + 0.0000i 760 −1 −0.0000 + 1.0000i 1−0.0000 − 1.0000i 761 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 762 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 763 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 764 −1 −0.0000 + 1.0000i 1 −0.0000 − 1.0000i 765 1  1.0000 −0.0000i 1  1.0000 + 0.0000i 766 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i767 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 768 1  0.0000 − 1.0000i −1 0.0000 + 1.0000i 769 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 770 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 771 −1  1.0000 + 0.0000i −1 1.0000 − 0.0000i 772 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 773 −1−1.0000 + 0.0000i −1 −1.0000 − 0.0000i 774 −1 −0.0000 − 1.0000i 1−0.0000 + 1.0000i 775 −1  1.0000 + 0.0000i −1  1.0000 − 0.0000i 776 −1 0.0000 + 1.0000i 1  0.0000 − 1.0000i 777 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 778 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 779 −1 1.0000 + 0.0000i −1  1.0000 − 0.0000i 780 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 781 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 782 −1 0.0000 − 1.0000i 1  0.0000 + 1.0000i 783 −1  1.0000 − 0.0000i −1 1.0000 + 0.0000i 784 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 785 1 1.0000 − 0.0000i 1  1.0000 + 0.0000i 786 1 −0.0000 + 1.0000i −1 −0.0000− 1.0000i 787 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 788 −1  0.0000 +1.0000i 1  0.0000 − 1.0000i 789 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i790 1 −0.0000 + 1.0000i −1 −0.0000 − 1.0000i 791 1 −1.0000 + 0.0000i 1−1.0000 − 0.0000i 792 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 793 −1−1.0000 − 0.0000i −1 −1.0000 + 0.0000i 794 1  0.0000 + 1.0000i −1 0.0000 − 1.0000i 795 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 796 1−0.0000 − 1.0000i −1 −0.0000 + 1.0000i 797 −1 −1.0000 − 0.0000i −1−1.0000 + 0.0000i 798 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 799 1−1.0000 + 0.0000i 1 −1.0000 − 0.0000i 800 −1  0.0000 + 1.0000i 1  0.0000− 1.0000i 801 −1 −1.0000 − 0.0000i −1 −1.0000 + 0.0000i 802 −1 −0.0000 −1.0000i 1 −0.0000 + 1.0000i 803 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i804 1  0.0000 − 1.0000i −1  0.0000 + 1.0000i 805 1  1.0000 − 0.0000i 1 1.0000 + 0.0000i 806 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 807 1−1.0000 + 0.0000i 1 −1.0000 − 0.0000i 808 1  0.0000 − 1.0000i −1 0.0000 + 1.0000i 809 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 810 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 811 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 812 −1 −0.0000 + 1.0000i 1 −0.0000 − 1.0000i 813 −1 −1.0000 +0.0000i −1 −1.0000 − 0.0000i 814 1  0.0000 + 1.0000i −1  0.0000 −1.0000i 815 1 −1.0000 − 0.0000i 1 −1.0000 + 0.0000i 816 −1  0.0000 +1.0000i 1  0.0000 − 1.0000i 817 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i818 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 819 −1  1.0000 + 0.0000i −1 1.0000 − 0.0000i 820 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 821 1 1.0000 − 0.0000i 1  1.0000 + 0.0000i 822 1  0.0000 + 1.0000i −1  0.0000− 1.0000i 823 1 −1.0000 − 0.0000i 1 −1.0000 + 0.0000i 824 1 −0.0000 −1.0000i −1 −0.0000 + 1.0000i 825 −1 −1.0000 + 0.0000i −1 −1.0000 −0.0000i 826 1 −0.0000 + 1.0000i −1 −0.0000 − 1.0000i 827 −1  1.0000 −0.0000i −1  1.0000 + 0.0000i 828 1 −0.0000 − 1.0000i −1 −0.0000 +1.0000i 829 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 830 1 −0.0000 +1.0000i −1 −0.0000 − 1.0000i 831 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i832 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 833 −1 −1.0000 − 0.0000i −1−1.0000 + 0.0000i 834 −1  0.0000 − 1.0000i 1  0.0000 + 1.0000i 835 1−1.0000 + 0.0000i 1 −1.0000 − 0.0000i 836 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 837 1  1.0000 + 0.0000i 1  1.0000 − 0.0000i 838 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 839 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 840 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 841 1  1.0000 +0.0000i 1  1.0000 − 0.0000i 842 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i843 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 844 −1 −0.0000 + 1.0000i 1−0.0000 − 1.0000i 845 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 846 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 847 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 848 −1 −0.0000 + 1.0000i 1 −0.0000 − 1.0000i 849 −1 −1.0000 +0.0000i −1 −1.0000 − 0.0000i 850 −1 −0.0000 − 1.0000i 1 −0.0000 +1.0000i 851 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 852 1  0.0000 −1.0000i −1  0.0000 + 1.0000i 853 −1 −1.0000 + 0.0000i −1 −1.0000 −0.0000i 854 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 855 −1  1.0000 +0.0000i −1  1.0000 − 0.0000i 856 −1  0.0000 + 1.0000i 1  0.0000 −1.0000i 857 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 858 −1 −0.0000 −1.0000i 1 −0.0000 + 1.0000i 859 1 −1.0000 − 0.0000i 1 −1.0000 + 0.0000i860 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 861 1  1.0000 − 0.0000i 1 1.0000 + 0.0000i 862 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 863 −1 1.0000 + 0.0000i −1  1.0000 − 0.0000i 864 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 865 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 866 −1 0.0000 − 1.0000i 1  0.0000 + 1.0000i 867 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 868 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 869 1  1.0000 −0.0000i 1  1.0000 + 0.0000i 870 1 −0.0000 + 1.0000i −1 −0.0000 − 1.0000i871 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 872 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 873 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 874 −1 0.0000 − 1.0000i 1  0.0000 + 1.0000i 875 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 876 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 877 −1 −1.0000 −0.0000i −1 −1.0000 + 0.0000i 878 1  0.0000 + 1.0000i −1  0.0000 −1.0000i 879 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 880 −1  0.0000 +1.0000i 1  0.0000 − 1.0000i 881 1  1.0000 + 0.0000i 1  1.0000 − 0.0000i882 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 883 −1  1.0000 − 0.0000i −1 1.0000 + 0.0000i 884 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 885 1 1.0000 + 0.0000i 1  1.0000 − 0.0000i 886 1  0.0000 + 1.0000i −1  0.0000− 1.0000i 887 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 888 1  0.0000 −1.0000i −1  0.0000 + 1.0000i 889 −1 −1.0000 + 0.0000i −1 −1.0000 −0.0000i 890 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 891 −1  1.0000 −0.0000i −1  1.0000 + 0.0000i 892 1  0.0000 − 1.0000i −1  0.0000 +1.0000i 893 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 894 1  0.0000 +1.0000i −1  0.0000 − 1.0000i 895 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i896 −1 −0.0000 + 1.0000i 1 −0.0000 − 1.0000i 897 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 898 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 899 1−1.0000 − 0.0000i 1 −1.0000 + 0.0000i 900 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 901 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 902 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 903 1 −1.0000 − 0.0000i 1−1.0000 + 0.0000i 904 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 905 1 1.0000 − 0.0000i 1  1.0000 + 0.0000i 906 −1 −0.0000 − 1.0000i 1−0.0000 + 1.0000i 907 1 −1.0000 − 0.0000i 1 −1.0000 + 0.0000i 908 −1 0.0000 + 1.0000i 1  0.0000 − 1.0000i 909 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 910 1 −0.0000 + 1.0000i −1 −0.0000 − 1.0000i 911 1−1.0000 + 0.0000i 1 −1.0000 − 0.0000i 912 −1  0.0000 + 1.0000i 1  0.0000− 1.0000i 913 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 914 −1  0.0000 −1.0000i 1  0.0000 + 1.0000i 915 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i916 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 917 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 918 −1  0.0000 − 1.0000i 1  0.0000 + 1.0000i 919 −1 1.0000 − 0.0000i −1  1.0000 + 0.0000i 920 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 921 1  1.0000 + 0.0000i 1  1.0000 − 0.0000i 922 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 923 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 924 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 925 1  1.0000 +0.0000i 1  1.0000 − 0.0000i 926 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i927 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 928 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 929 1  1.0000 + 0.0000i 1  1.0000 − 0.0000i 930 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 931 −1  1.0000 − 0.0000i −1 1.0000 + 0.0000i 932 −1 −0.0000 + 1.0000i 1 −0.0000 − 1.0000i 933 −1−1.0000 + 0.0000i −1 −1.0000 − 0.0000i 934 −1 −0.0000 − 1.0000i 1−0.0000 + 1.0000i 935 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 936 −1−0.0000 + 1.0000i 1 −0.0000 − 1.0000i 937 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 938 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 939 −1 1.0000 − 0.0000i −1  1.0000 + 0.0000i 940 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 941 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 942 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 943 −1  1.0000 + 0.0000i −1 1.0000 − 0.0000i 944 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 945 −1−1.0000 + 0.0000i −1 −1.0000 − 0.0000i 946 −1 −0.0000 − 1.0000i 1−0.0000 + 1.0000i 947 1 −1.0000 − 0.0000i 1 −1.0000 + 0.0000i 948 1−0.0000 − 1.0000i −1 −0.0000 + 1.0000i 949 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 950 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 951 −1 1.0000 − 0.0000i −1  1.0000 + 0.0000i 952 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 953 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 954 −1 0.0000 − 1.0000i 1  0.0000 + 1.0000i 955 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 956 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 957 1  1.0000 −0.0000i 1  1.0000 + 0.0000i 958 −1  0.0000 − 1.0000i 1  0.0000 + 1.0000i959 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 960 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 961 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 962 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 963 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 964 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 965 1  1.0000 +0.0000i 1  1.0000 − 0.0000i 966 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i967 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 968 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 969 1  1.0000 + 0.0000i 1  1.0000 − 0.0000i 970 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 971 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 972 −1 −0.0000 + 1.0000i 1 −0.0000 − 1.0000i 973 −1 −1.0000 +0.0000i −1 −1.0000 − 0.0000i 974 1  0.0000 + 1.0000i −1  0.0000 −1.0000i 975 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 976 −1 −0.0000 +1.0000i 1 −0.0000 − 1.0000i 977 −1 −1.0000 + 0.0000i −1 −1.0000 −0.0000i 978 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 979 1 −1.0000 +0.0000i 1 −1.0000 − 0.0000i 980 1  0.0000 − 1.0000i −1  0.0000 + 1.0000i981 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 982 −1 −0.0000 − 1.0000i 1−0.0000 + 1.0000i 983 −1  1.0000 + 0.0000i −1  1.0000 − 0.0000i 984 −1 0.0000 + 1.0000i 1  0.0000 − 1.0000i 985 1  1.0000 − 0.0000i 1 1.0000 + 0.0000i 986 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 987 1−1.0000 − 0.0000i 1 −1.0000 + 0.0000i 988 −1  0.0000 + 1.0000i 1  0.0000− 1.0000i 989 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 990 −1 −0.0000 −1.0000i 1 −0.0000 + 1.0000i 991 −1  1.0000 + 0.0000i −1  1.0000 −0.0000i 992 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 993 −1 −1.0000 +0.0000i −1 −1.0000 − 0.0000i 994 −1  0.0000 − 1.0000i 1  0.0000 +1.0000i 995 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 996 1 −0.0000 −1.0000i −1 −0.0000 + 1.0000i 997 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i998 1 −0.0000 + 1.0000i −1 −0.0000 − 1.0000i 999 1 −1.0000 + 0.0000i 1−1.0000 − 0.0000i 1000 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 1001 1 1.0000 − 0.0000i 1  1.0000 + 0.0000i 1002 −1  0.0000 − 1.0000i 1 0.0000 + 1.0000i 1003 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 1004 −1 0.0000 + 1.0000i 1  0.0000 − 1.0000i 1005 −1 −1.0000 − 0.0000i −1−1.0000 + 0.0000i 1006 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 1007 1−1.0000 + 0.0000i 1 −1.0000 − 0.0000i 1008 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 1009 1  1.0000 + 0.0000i 1  1.0000 − 0.0000i 1010 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 1011 −1  1.0000 − 0.0000i −1 1.0000 + 0.0000i 1012 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 1013 1 1.0000 + 0.0000i 1  1.0000 − 0.0000i 1014 1  0.0000 + 1.0000i −1 0.0000 − 1.0000i 1015 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 1016 1 0.0000 − 1.0000i −1  0.0000 + 1.0000i 1017 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 1018 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 1019 −1 1.0000 − 0.0000i −1  1.0000 + 0.0000i 1020 1  0.0000 − 1.0000i −1 0.0000 + 1.0000i 1021 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 1022 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 1023 1 −1.0000 + 0.0000i 1−1.0000 − 0.0000i 1024 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 1025 1 1.0000 − 0.0000i 1  1.0000 + 0.0000i 1026 1  0.0000 + 1.0000i −1 0.0000 − 1.0000i 1027 −1  1.0000 + 0.0000i −1  1.0000 − 0.0000i 1028 −1 0.0000 + 1.0000i 1  0.0000 − 1.0000i 1029 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 1030 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 1031 −1 1.0000 + 0.0000i −1  1.0000 − 0.0000i 1032 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 1033 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 1034 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 1035 −1  1.0000 − 0.0000i −1 1.0000 + 0.0000i 1036 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 1037 1 1.0000 − 0.0000i 1  1.0000 + 0.0000i 1038 −1  0.0000 − 1.0000i 1 0.0000 + 1.0000i 1039 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 1040 1−0.0000 − 1.0000i −1 −0.0000 + 1.0000i 1041 1  1.0000 − 0.0000i 1 1.0000 + 0.0000i 1042 1 −0.0000 + 1.0000i −1 −0.0000 − 1.0000i 1043 −1 1.0000 − 0.0000i −1  1.0000 + 0.0000i 1044 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 1045 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 1046 1 0.0000 + 1.0000i −1  0.0000 − 1.0000i 1047 1 −1.0000 + 0.0000i 1−1.0000 − 0.0000i 1048 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 1049 −1−1.0000 − 0.0000i −1 −1.0000 + 0.0000i 1050 1  0.0000 + 1.0000i −1 0.0000 − 1.0000i 1051 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 1052 1−0.0000 − 1.0000i −1 −0.0000 + 1.0000i 1053 −1 −1.0000 − 0.0000i −1−1.0000 + 0.0000i 1054 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 1055 1−1.0000 + 0.0000i 1 −1.0000 − 0.0000i 1056 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 1057 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 1058 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 1059 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 1060 1  0.0000 − 1.0000i −1  0.0000 + 1.0000i 1061 1  1.0000 −0.0000i 1  1.0000 + 0.0000i 1062 1  0.0000 + 1.0000i −1  0.0000 −1.0000i 1063 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 1064 1  0.0000 −1.0000i −1  0.0000 + 1.0000i 1065 1  1.0000 − 0.0000i 1  1.0000 +0.0000i 1066 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 1067 1 −1.0000 +0.0000i 1 −1.0000 − 0.0000i 1068 −1  0.0000 + 1.0000i 1  0.0000 −1.0000i 1069 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 1070 1  0.0000 +1.0000i −1  0.0000 − 1.0000i 1071 1 −1.0000 − 0.0000i 1 −1.0000 +0.0000i 1072 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 1073 1  1.0000 −0.0000i 1  1.0000 + 0.0000i 1074 1  0.0000 + 1.0000i −1  0.0000 −1.0000i 1075 −1  1.0000 + 0.0000i −1  1.0000 − 0.0000i 1076 −1  0.0000 +1.0000i 1  0.0000 − 1.0000i 1077 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i1078 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 1079 1 −1.0000 + 0.0000i 1−1.0000 − 0.0000i 1080 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 1081 −1−1.0000 + 0.0000i −1 −1.0000 − 0.0000i 1082 1 −0.0000 + 1.0000i −1−0.0000 − 1.0000i 1083 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 1084 1−0.0000 − 1.0000i −1 −0.0000 + 1.0000i 1085 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 1086 1 −0.0000 + 1.0000i −1 −0.0000 − 1.0000i 1087 1−1.0000 + 0.0000i 1 −1.0000 − 0.0000i 1088 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i 1089 −1 −1.0000 − 0.0000i −1 −1.0000 + 0.0000i 1090 −1−0.0000 − 1.0000i 1 −0.0000 + 1.0000i 1091 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 1092 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 1093 1  1.0000 +0.0000i 1  1.0000 − 0.0000i 1094 1  0.0000 + 1.0000i −1  0.0000 −1.0000i 1095 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 1096 1 −0.0000 −1.0000i −1 −0.0000 + 1.0000i 1097 1  1.0000 + 0.0000i 1  1.0000 −0.0000i 1098 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 1099 1 −1.0000 +0.0000i 1 −1.0000 − 0.0000i 1100 −1 −0.0000 + 1.0000i 1 −0.0000 −1.0000i 1101 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 1102 1  0.0000 +1.0000i −1  0.0000 − 1.0000i 1103 1 −1.0000 + 0.0000i 1 −1.0000 −0.0000i 1104 −1 −0.0000 + 1.0000i 1 −0.0000 − 1.0000i 1105 −1 −1.0000 +0.0000i −1 −1.0000 − 0.0000i 1106 −1 −0.0000 − 1.0000i 1 −0.0000 +1.0000i 1107 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 1108 1  0.0000 −1.0000i −1  0.0000 + 1.0000i 1109 −1 −1.0000 + 0.0000i −1 −1.0000 −0.0000i 1110 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 1111 −1  1.0000 +0.0000i −1  1.0000 − 0.0000i 1112 −1  0.0000 + 1.0000i 1  0.0000 −1.0000i 1113 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i 1114 −1 −0.0000 −1.0000i 1 −0.0000 + 1.0000i 1115 1 −1.0000 − 0.0000i 1 −1.0000 + 0.0000i1116 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 1117 1  1.0000 − 0.0000i 1 1.0000 + 0.0000i 1118 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 1119 −1 1.0000 − 0.0000i −1  1.0000 + 0.0000i 1120 1 −0.0000 − 1.0000i −1−0.0000 + 1.0000i 1121 −1 −1.0000 + 0.0000i −1 −1.0000 − 0.0000i 1122 −1 0.0000 − 1.0000i 1  0.0000 + 1.0000i 1123 1 −1.0000 + 0.0000i 1 −1.0000− 0.0000i 1124 1 −0.0000 − 1.0000i −1 −0.0000 + 1.0000i 1125 1  1.0000 −0.0000i 1  1.0000 + 0.0000i 1126 1 −0.0000 + 1.0000i −1 −0.0000 −1.0000i 1127 1 −1.0000 + 0.0000i 1 −1.0000 − 0.0000i 1128 1 −0.0000 −1.0000i −1 −0.0000 + 1.0000i 1129 1  1.0000 − 0.0000i 1  1.0000 +0.0000i 1130 −1 −0.0000 − 1.0000i 1 −0.0000 + 1.0000i 1131 1 −1.0000 +0.0000i 1 −1.0000 − 0.0000i 1132 −1  0.0000 + 1.0000i 1  0.0000 −1.0000i 1133 −1 −1.0000 − 0.0000i −1 −1.0000 + 0.0000i 1134 1  0.0000 +1.0000i −1  0.0000 − 1.0000i 1135 1 −1.0000 + 0.0000i 1 −1.0000 −0.0000i 1136 −1  0.0000 + 1.0000i 1  0.0000 − 1.0000i 1137 1  1.0000 +0.0000i 1  1.0000 − 0.0000i 1138 1  0.0000 + 1.0000i −1  0.0000 −1.0000i 1139 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 1140 −1  0.0000 +1.0000i 1  0.0000 − 1.0000i 1141 1  1.0000 − 0.0000i 1  1.0000 + 0.0000i1142 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 1143 1 −1.0000 + 0.0000i 1−1.0000 − 0.0000i 1144 1  0.0000 − 1.0000i −1  0.0000 + 1.0000i 1145 −1−1.0000 + 0.0000i −1 −1.0000 − 0.0000i 1146 1  0.0000 + 1.0000i −1 0.0000 − 1.0000i 1147 −1  1.0000 − 0.0000i −1  1.0000 + 0.0000i 1148 1 0.0000 − 1.0000i −1  0.0000 + 1.0000i 1149 −1 −1.0000 + 0.0000i −1−1.0000 − 0.0000i 1150 1  0.0000 + 1.0000i −1  0.0000 − 1.0000i 1151 1−1.0000 + 0.0000i 1 −1.0000 − 0.0000i 1152 −1  0.0000 + 1.0000i 1 0.0000 − 1.0000i

What is claimed is:
 1. A method performed by a wireless transmit/receiveunit (WTRU), the method comprising: determining a number of space-timestreams for a Multi-Input Multi-Output (MIMO) transmission; generatingthe MIMO transmission for the number of space-time streams, wherein theMIMO transmission comprises at least one channel estimation field (CEF)for each respective space-time stream, wherein the number of CEFs foreach respective space-time stream in the MIMO transmission is based on amatrix associated with the number of space-time-streams, and wherein thenumber of CEFs for each said space-time stream in the MIMO transmissionis less than the total number of space-time streams in the MIMOtransmission; wherein a first space-time stream is orthogonal to atleast one second space-time stream, and wherein each CEF of the firstspace-time stream has a corresponding CEF of the at least one secondspace-time stream that is based on a same value in the matrix; andtransmitting the MIMO transmission comprising the number of CEFs foreach space-time stream.
 2. The method of claim 1, wherein the firstspace-time stream includes information associated with a first set ofcomplex numbers, and the second space-time stream includes informationassociated with a second set of complex numbers.
 3. The method of claim2, wherein the second set of complex numbers are complex conjugates ofthe first set of complex numbers.
 4. The method of claim 3, wherein theMIMO transmission comprises a physical layer (PHY) frame that includessimultaneous transmission of the first space-time stream including thefirst set of complex numbers and the second space-time stream includingthe second set of complex numbers.
 5. The method of claim 1, wherein theorthogonality between the first space-stream and the second space streamis maintained by the matrix.
 6. The method of claim 1, wherein thenumber of CEFs for each space-time stream in the MIMO transmission ishalf the number of space-time streams.
 7. The method of claim 1, whereineach space-time stream is transmitted on a respective antenna.
 8. Awireless transmit/receive unit (WTRU) comprising: a processor configuredto: determine a number of space-time streams for a Multi-InputMulti-Output (MIMO) transmission; generate the MIMO transmission for thenumber of space-time streams, wherein the MIMO transmission comprises atleast one channel estimation field (CEF) for each respective space-timestream, wherein the number of CEFs for each respective space-time streamin the MIMO transmission is based on a matrix associated with the numberof space-time streams, and wherein the number of CEFs for eachrespective space-time stream in the MIMO transmission is less than thetotal number of space-time streams in the MIMO transmission; wherein afirst space-time stream is orthogonal to at least one second space-timestream, and wherein each CEF of the first space-time stream has acorresponding CEF of the at least one second space-time stream that isbased on a same value in the matrix; and transmit the MIMO transmissioncomprising the number of CEFs for each space-time stream.
 9. The WTRU ofclaim 8, wherein the first space-time stream includes informationassociated with a first set of complex numbers, and the secondspace-time stream includes information associated with a second set ofcomplex numbers.
 10. The WTRU of claim 9, wherein the second set ofcomplex numbers are complex conjugates of the first set of complexnumbers.
 11. The WTRU of claim 10, wherein the MIMO transmissioncomprises a physical layer (PHY) frame that includes simultaneoustransmission of the first space-time stream including the first set ofcomplex numbers and the second space-time stream including the secondset of complex numbers.
 12. The WTRU of claim 8, wherein theorthogonality between the first space-stream and the second space streamis maintained by the matrix.
 13. The WTRU of claim 8, wherein the numberof CEFs for each space-time stream in the MIMO transmission is half thenumber of space-time streams.
 14. The WTRU of claim 8, wherein eachspace-time stream is transmitted on a respective antenna.